Grace–van Zwam's frame-template structure conjecture

Let $\mathbb F$ be a finite field, let $m$ be a positive integer, and let $\mathcal M$ be a minor-closed class of $\mathbb F$-represented matroids. A frame template specifies a class $\mathcal{M}(\Phi)$ of represented matroids conforming to it, and $\widetilde M$ denotes the underlying matroid of a represented matroid $M$; when $\mathbb F$ has characteristic $p\ne0$, $\mathbb F_p$ is its prime subfield.

Frame-template conjecture. There exist $k\in\mathbb Z_+$ and frame templates $\Phi_1,\ldots,\Phi_s,\Psi_1,\ldots,\Psi_t$ such that $\mathcal{M}$ contains each class $\mathcal{M}(\Phi_i)$, contains the duals of the represented matroids in each class $\mathcal{M}(\Psi_j)$, and, whenever $M$ is a simple vertically $k$-connected member of $\mathcal M$ whose underlying matroid $\widetilde M$ has no $PG(m-1,\mathbb F_p)$-minor, either $M$ belongs to some $\mathcal{M}(\Phi_i)$ or $M^*$ belongs to some $\mathcal{M}(\Psi_j)$.

The template formulation is intended as a weaker replacement for the perturbation conjecture and to describe highly connected members of minor-closed classes. The paper explicitly states that this conjecture is false, so it is refuted.