Geelen–Gerards–Whittle conjecture on branch-width under circuit-hyperplane relaxation

Let $M_1$ be a matroid with a circuit-hyperplane $H$, and let $M_2$ be obtained from $M_1$ by relaxing $H$. If both $M_1$ and $M_2$ are representable over a finite field $\mathbb F$, then the branch-width of $M_1$ should be bounded by a constant depending only upon $|\mathbb F|$. This conjecture is the second conjecture used in the reported proof of Rota's Conjecture, and the paper's main result shows that it implies the corresponding fragile-matroid branch-width conjecture. The source reports that its proof was also reported by Geelen, Gerards, and Whittle.