Weighted fractional matching-cover conjecture for matroid intersections

Let $\mathcal{L}\in\mathcal{D}^k$ be a family of $k$ matroids on $V$, and let $w\in\mathbb{R}_{\ge0}^V$. Weighted fractional matching-cover conjecture. One has

$$\tau_w^*(\mathcal{L})\le (k-1)\nu_w(\mathcal{L}).$$

This is presented as the weighted generalization of the integral matching-cover gap conjecture. It is proved in the paper for intersections of partition matroids, while the general case remains open.