Weighted exchange-distance conjecture for matroid basis pairs

Let $M$ be a matroid over a ground set $S$, let $\mathbf{P}_1=(R_1,B_1)$ and $\mathbf{P}_2=(R_2,B_2)$ be compatible pairs of disjoint bases, and let $w\colon S\to\mathbb{R}_+$ be a weight function. The weighted exchange distance is the minimum total weight of symmetric exchanges transforming one pair into the other, or $+\infty$ if no such sequence exists. For a set $A\subseteq S$, write $w(A)$ for the sum of the weights of its elements.

Weighted exchange-distance conjecture. The weighted exchange distance of $\mathbf{P}_1$ and $\mathbf{P}_2$ is at most

$$w(R_1\cup B_1)=w(R_2\cup B_2).$$

The paper proposes this as a weighted extension of Hamidoune's conjecture and proves it for several classes of matroids, including strongly base orderable, split, and certain graphic matroids, while the general case remains open.