Rota's excluded-minor finiteness conjecture for representable matroids

Let $F$ be a finite field, and consider the class of matroids representable over $F$. An excluded minor for this class is a matroid that is not $F$-representable, while every proper minor is $F$-representable.

Rota's conjecture. For each finite field $F$, there are, up to isomorphism, only finitely many excluded minors for the class of $F$-representable matroids.

This is a central finiteness conjecture in matroid minor theory. The supplied text states it as a conjecture and gives no resolution or status evidence.