Weighted cyclic-exchange conjecture for basis sequences

Let $(B_1,\dots,B_k)$ and $(B'_1,\dots,B'_k)$ be compatible sequences of $k$ bases of a matroid $M=(S,\mathcal{B})$, and let $w:S\to\mathbb{R}_+$. Weighted cyclic-exchange conjecture. The sequence $(B_1,\dots,B_k)$ can be transformed into $(B'_1,\dots,B'_k)$ by cyclic exchanges of total weight at most

$$\frac{1}{k}\sum_{i=1}^k w(B_i)=\frac{1}{k}\sum_{i=1}^k w(B'_i).$$

This is the weighted counterpart of the bounded cyclic-exchange conjecture and is presented as open in the source.