The partial-field inequivalent-representation chain conjecture

Let $\mathbb{P}$ be a partial field, and let $\operatorname{\mathcal{M}}(\mathbb{P})$ denote the class of $\mathbb{P}$-representable matroids. Partial-field inequivalent-representation chain conjecture. If $\operatorname{\mathcal{M}}(\mathbb{P})$ has infinitely many excluded minors, then there is an infinite chain of matroids $N_1,N_2,\ldots$ such that $N_i$ has at least $i$ inequivalent representations over $\mathbb{P}$ and $N_i$ is a minor of some excluded minor. The conjecture would relate an infinite excluded-minor obstruction to unboundedly many inequivalent partial-field representations; the source presents it as an unresolved conjecture closely related to its main theorem.