Finite-field list conjecture for matroid representability

Let $k$ be a prime power. The claim concerns matroids and representability over fields and finite fields $\operatorname{GF}(q)$. Finite-field list conjecture. There exists a number $n_k$ such that, for all matroids $M$, $M$ is representable over all fields with at least $k$ elements if and only if it is representable over all finite fields $\operatorname{GF}(q)$ with $k \leq q \leq n_k$. This generalizes the characterization of near-regular matroids; the paper states that the conjecture remains open, including for $k=4$.