The idempotency conjecture for finitely generated parasemifields

A parasemifield is a semiring in which every nonzero element has a multiplicative inverse. A parasemifield is finitely generated as a semiring if it is generated by finitely many elements under addition and multiplication. It is additively idempotent when

$$a+a=a$$

for every element $a$.

Idempotency conjecture. Every parasemifield which is finitely generated as a semiring is additively idempotent.

There are no finite parasemifields, and the conjecture asks for the structure of finitely generated parasemifields without assuming additive idempotency. It is motivated by the analogous fact that a field finitely generated as a ring is finite; its status is not determined by the supplied text.