The high-branch-width non-fragility conjecture

Let $N$ be a $\operatorname{GF}(q)$-representable matroid, and let $\operatorname{bw}(M)$ denote the branch width of a matroid $M$. High-branch-width non-fragility conjecture. There is an integer $l$, depending only on $N$ and $q$, such that, if $M$ is a $\operatorname{GF}(q)$-representable matroid with $\operatorname{bw}(M)>l$ and $N$ is a minor of $M$, then there exists an element $e\in E(M)$ for which both $M{-} e$ and $M/e$ have a minor isomorphic to $N$. This is a weaker version of a conjecture attributed in the source to Geelen et al.; the stronger version requires the same fixed $N$-minor after both operations, and the present conjecture remains open in the source.