Most of my sporadic work on this TAS has been staring at the 2003- and 2004-turn plans; I think we're agreed that the current 2003-turn plan is unlikely to ever happen, but that doesn't rule out the possbility of a different 2003-turn plan that works.
So far, though, all my attempts (that don't do a hypercharged crossing of Air, the hardest part) fall one action short. Here's a rundown of the problem.
First off, a fundamental rule of NetHack is that if you change levels on a monster action, no monster actions happen when you arrive, not even ones later in the turn; either you get a player action (if you have the movement points for it), or a turn boundary (if you don't). (Incidentally, this is what causes the hypercharge glitch.) This places a hard "speed limit" on how fast we can move; we can change levels once on our action, once on the monster action, and then we have to spend another action in order to keep moving (theoretically there's the alternative of having < 12 movement points in a turn, but that's clearly suboptimal to just using an action).
In 2003 turns, we have 5+4+4+4 = 17 actions (air elemental) or 5+4+5+4 = 18 actions (riding, requires grinding, also requires a "maintain gallop" player move on two specific actions that mean we only really have 16 usable player actions) in the fastest available movement forms. Thus, an "idealistic" plan that glosses over details and assumes things to be possible that probably aren't looks like this:
Action 1: gain alignment; get accepted to Quest; levport to Quest goal level (STOP; we can't take three actions on our own action, thus at least one of those was a monster action and we changed level)
Action 2: get Eye, Bell (…no further use for this action, using the Eye is the fastest way to leave the Quest and it can't happen on a monster turn; if a monster tried to help us out, we'd change level in the process, the action would end, and using the Eye would still be faster)
Action 3: branchport to Valley or Vlad's (…no further use for this action, monsters can't cause us to leave Vlad's or change levels from the Valley, unless I'm missing something)
Actions 4, 5, 6, 7: levport to Vibrating Square, ring Bell, read Book (×2) (all of these need to happen as player actions); we can enter the Sanctum on the subsequent monster action (STOP; changed level on a monster action)
Action 8: retrieve Amulet (this can happen on a player or monster action, but can't happen on action 7's monster action because you don't get monster actions after a level change; and we can't leave the Sanctum on a monster action as there are no levport traps or holes there)
Action 9: leave Sanctum (the fastest known glitch for fast Amulet transport cannot be done there); fast Amulet transport to Dlvl 1 (STOP; we changed level twice in an action, thus one was a monster action)
Action 10: enter Earth (…no further use for this action? enexto is the current fastest method of crossing Earth and requires a player action to set up, but maybe there are others; I've considered hypercharged dig + horizontal movement but it seems to be slower)
Action 11: #monster; enexto to Air (STOP; we entered Air on a monster action, because that's how the enexto glitch works)
Action 12: overflow hurtle to Fire (thanks to whoever found this glitch; sadly I've forgotten who it was, but the idea is that kicking off a monster interacts in confusing but sometimes beneficial ways with antigold); hurtle recovery via equipment storage theft (this is a recent discovery of mine: if you're interrupted removing an item via A, avoid performing various actions, and then the item gets stolen, it cancels helplessness); any monster actions useful here? (Note that if you want to do more than a trivial number of monster actions, you'll have to explain how a highly charged monster ended up on Fire. This is a theoretical plan that ignores that sort of issue in case we find a way; currently it seems like it might be just barely possible to get a monster there with a charge of 1, and even that involves chameleons mimicking air elementals and polymorphing at just the right moment.)
Action 13: enter Water, somehow; enter Astral, somehow (!), STOP because we changed level twice
Action 14: #monster; enexto to altar
Action 15: #offer
It might look like we have time to spare. The problem is that some of the above steps are overly idealistic, in particular, the actions entering Water and following that. The fastest known strategy for entering Water onwards looks like this:
2002 (1 action remaining): jump into the portal to Water {requires: air-E form for the 4-action turn, wearing Eyes of the Overworld for the ability to jump as an air-E}
2003 (turn boundary): polymorph to p/t form, fill level with eggs
2003 (4 actions remaining): #monster; enexto into portal
2003 (3 actions remaining): zap turn undead, filling the level with monsters
2003 (2 actions remaining): #monster; enexto onto altar (yet to be figured out: where the trigger monster came from and how it got charged to 3; probably we'd need a second leashed monster for the purpose)
2003 (1 action remaining): #offer and win
That's two actions slower than the excessively idealistic plan, bringing it up to 17. So we have to somehow make our idealistic plan into a real plan without spending any extra actions. The main blocking problem is polyform management; we need to be an air-E at each turn boundary, but an air-E can't read the Book or jump when blind, and can't use polytraps or levport traps when nonblind (because the Eyes of the Overworld give MR). Levport traps are crucial to the run (being required for the fastest version of the fast amulet transport glitch), which basically implies that we don't have the Eyes in inventory on turn 2001, and thus aren't in air-E form through most of that turn. (We can turn back in zero time, and in time for the turn 2002 boundary, via hitting a polymorph trap while being teleported around.) The problem is turn 2002; we somehow need to change from p/t form (used for the enexto on Earth) into air-E form, and also put on the Eyes (which we can't have had in inventory on turn 2001, thus can't have been wearing). One of those actions can be "oLS-cancelled, but the other looks like it has to be spent; there are no polytraps on Air or Fire. And unfortunately, that puts us one action over.
We also need to find a way to set up equipment storage theft: the setup (i.e. the "storage" half) takes one player action, apparently no matter what you do ("oLS-cancelling it doesn't work, not to mention that we've run out of "oLS-cancels already). The storage is broken if you move using a movement command, get engulfed, drop an item intentionally, or change equipment (intentionally or unintentionally). Getting a 5-action turn as turn 2000 requires the use of a movement command, so we can't set it up in advance; however, we can set it up while in the Sanctum, as we have a spare player action there (we can let monsters do the Amulet retrieval). So this takes some thought, but isn't a blocker.
One assumption I made above is that changing level always consumes a player action or monster action. One hope for a 2003-turn run would be that this assumption is not correct. While writing this post, I had the idea of "oLS-cancelling a levport, and found a way to do it: you can blast yourself to death with the Orb of Fate without violating any of the other restrictions on "oLS-cancelling. This also cancels the subsequent monster turn, but there are nonetheless actions (such as 4 in the above plan) that can be deleted via this method without losing time. The problem is that putting on a "oLS (amulet of life saving) costs one player action, and "oLS-cancelling saves one player action, so you can only gain time from it once; and we're already spending that gain elsewhere. Changing the timing of turn boundaries doesn't seem obviously helpful for this purpose.