Bérczi–Mátravölgyi–Schwarcz weighted symmetric-exchange conjecture

Let $(B_1,B_2)$ and $(B'_1,B'_2)$ be compatible basis pairs of a matroid $M=(S,\mathcal{B})$, and let $w:S\to\mathbb{R}_+$. The weight of a symmetric exchange replacing $B_1,B_2$ by $B_1-e+f,B_2-f+e$ is $w(e)/2+w(f)/2$. Bérczi–Mátravölgyi–Schwarcz conjecture. The pair $(B_1,B_2)$ can be transformed into $(B'_1,B'_2)$ by symmetric exchanges of total weight at most

$$w(B_1)/2+w(B_2)/2=w(B'_1)/2+w(B'_2)/2.$$

Setting all weights equal to $1$ recovers Hamidoune's conjecture; the weighted assertion is presented as open in the source.