Geelen–Gerards–Whittle's perturbation conjecture for represented matroids

Let $\mathbb F$ be a finite field and let $\mathcal M$ be a proper minor-closed class of $\mathbb F$-represented matroids. An $\mathbb F$-represented matroid is a matroid with a fixed representation over $\mathbb F$; a represented frame matroid has a representation matrix with at most two nonzero entries per column; and a rank-$(\leq t)$ perturbation is obtained by adding a matrix of rank at most $t$ to a representation matrix.

Perturbation conjecture. There exist $k,t\in\mathbb Z_+$ such that each vertically $k$-connected member of $\mathcal M$ is a rank-$(\leq t)$ perturbation of an $\mathbb F$-represented matroid $N$ such that either $N$ is a represented frame matroid, $N^*$ is a represented frame matroid, or $N$ is confined to a proper subfield of $\mathbb F$.

This is the perturbation form of the announced structural result for highly connected representable matroids. The paper states that its dyadic examples provide a counterexample, so the conjecture is refuted.