The well-quasi-ordering conjecture for finite-field-representable matroids

Let $\mathrm{GF}(q)$ be a finite field. A set of matroids is well-quasi-ordered by the minor relation when every infinite set contains two matroids, one of which is isomorphic to a minor of the other.

Well-quasi-ordering conjecture. Any infinite set of $\mathrm{GF}(q)$-representable matroids contains two matroids, one of which is isomorphic to a minor of the other.

A positive answer would imply that every minor-closed class of $\mathrm{GF}(q)$-representable matroids has finitely many $\mathrm{GF}(q)$-representable excluded minors. The conjecture is attributed to Neil Robertson and Paul Seymour and is discussed in the context of extending the Graph Minors project to matroids representable over finite fields; its resolution is not supplied here.