[RPG] Can others refluff their weapons in AL, or are “Unique Weapons” exclusive to Monks

As long as you use the stats of an existing weapon, is it legal to describe/name a weapon something else?

Monks are explicitly given this leeway (PHB page 79):

Certain monastaries use specialized forms of Monk weapons. For example, you might use a club that is two lengths of wood connected by a short chain (called a nunchaku) or a sickle with a shorter, straighter blade (called a kama). Whatever name you use for a mok weapon, you can use the game statistic provided for the weapon in chapter 5, 'Equipment.'

Adventurer's League officially condones a similar practice as far as races go (ALPGv4 page 3):

You can indicate that you are a member of a rare elf subrace for the purpose of roleplaying, but you can use only the subrace traits for elves given in the Player’s Handbook

So as long as the stats match an existing weapon (and you don't use the same name as an existing weapon, to limit confusion), is it legal for any player to do the same thing as Monks with any weapon?

My thinking is that in AL you will likely have many DMs, so while "DM decision" would be a nice answer, it can be frustrating to a player who has built a concept around a custom weapon (for example a barbarian who uses a greataxe that is restyled as a Soul Calibur style zweihander to set it apart from a regular greatsword) to have to ask at every session if it's "okay to use his imagination" so that his character isn't "Generic Barbarian #2" (which is inertia you already have to fight in AL games). I think in most private games the DM would shrug and say "sure," because refluffing is generally considered legal, but is it "official" enough to do in an AL game with at least some degree of confidence? I've met many AL DMs who have never even heard of refluffing and might reject it just out of uncertainty and not because it's against the rules.