Officially, No. Jeremy Crawford unofficially says that it should be based on actual stats, not temporary stats, but DMs are free to rule how they like.
However, in many House Rules, there is no minimum to LEAVE a class (i.e. profession), This is called "getting fired". Example of this, Head Priest says, "Your lack of wisdom shows us you aren't cut out for this son. Don't let the Temple doors hit your ass on the way out."
Many DMs find that minimum to leave is a stupid concept, and that too low of a base stat SHOULD get you punted from that class, rather than being trapped there forever. You should be unable to advance past where you are and forced out of the class until your stat changes for the better. Can you really justify trapping & mandating a 20th level High Priest with an 11 Wisdom score (if they survived)?
Advice if you choose to allow Temp stats to qualify. If they lose the item, they can't progress and should be ejected from the class. By RAW parallel game examples where PCs take permanent ability damage (from dozens of ways in the game), lowered stats does NOT nullify class powers/abilities or feats already acquired, only lowers their abilities based on the damaged stat accordingly, like spells known, save DC, attack bonus, relevant saves and related skills.
For an example, the PC would be they use the Headband to get in and out of Div. Wizard and say, back to BM Fighter. They lose the Headband two adventures later and their Int drops to its original 12. by RAW, they keep the Portent & Arcane Recovery abilities and their cantrip spells, but their spells known, spell attack bonus, Wizard Save DC, Int save and Int based skills all get lowered accordingly. For a slightly different example, if they were still in the Wizard class, the above would be the same, but they would be unable to advance any further in the Wizard class until their stat changed to 13 or higher. If a new level was acquired, they could still advance in the existing Fighter class, or any other class, that they qualify for.