The weighted bounded-exchange conjecture for F-avoiding matroid bases

Let $M$ be a matroid on a ground set $E$, let $\Gamma$ be an abelian group, let $\psi\colon E\to\Gamma$ be a group labeling, let $F\subseteq\Gamma$ be a finite set, and let $w\colon E\to\mathbb{R}$ be a weight function. An $F$-avoiding basis is a basis whose label is not in $F$. Weighted bounded-exchange conjecture. If $M$ has an $F$-avoiding basis, then for every minimum-weight basis $B$, there is a minimum-weight $F$-avoiding basis $B^*$ such that $|B\setminus B^*|\leq |F|$. This stronger weighted analogue is posed as an open problem alongside the unweighted conjectures.