The finitary partial-field excluded-minor conjecture

A partial field $\mathbb{P}$ is finitary if it admits a homomorphism to $\operatorname{GF}(q)$ for some prime power $q$. Let $\operatorname{\mathcal{M}}(\mathbb{P})$ denote the class of $\mathbb{P}$-representable matroids. Finitary partial-field excluded-minor conjecture. For every finitary partial field $\mathbb{P}$, the class $\operatorname{\mathcal{M}}(\mathbb{P})$ can be characterized by a finite set of excluded minors. This generalizes Rota's Conjecture from finite fields to finitary partial fields; the source says it has been settled for only a handful of partial fields.