Does Alchemist’s Fire damage get rolled once, or every turn?

Alchemist’s Fire does things quite differently from other items, being an improvised weapon that must always use ranged attacks, and deals its damage on subsequent turns, not when thrown. And Jeremy Crawford has tweeted a number of clarifications about how the damage works. But there’s one thing that I haven’t found any answers about, whether official, tweet, or even just accepted common practice. Does the damage get re-rolled every turn, or only the once?

On the one hand, only once would make sense with Crawford’s tweets; Dexterity gets added to the damage it deals, because that damage is the one roll made for the weapon’s damage, and the usual rule applies. Which feels more sensible than adding Dex to every roll, turn after turn.

On the other hand, there’s no rule to explicitly say that it’s rolled only once, and ongoing damage is usually re-rolled every time.

So, is the idea supposed to be that the improvised range weapon deals 1d4+dex damage, and the target takes that same damage every turn, or is it that they take a re-rolled 1d4+dex every turn?

For context: I DM, and a player with a Fast Hands-using Thief just brought this up. The whole Dex-to-ongoing-damage thing had always felt a bit off to both of us. He had an idea that his might be the intended interpretation, and I can’t actually find anything anywhere that actually definitively rules either way.