Weighted fractional coloring conjecture for intersections of matroids

Let $\mathcal{L}=\{\mathcal{M}_1,\ldots,\mathcal{M}_k\}\in\mathcal{D}^k$ be matroids on $V$, let $\mathcal{C}=\bigcap\mathcal{L}$, and let $\vec h\in\mathbb{R}_{\ge0}^V$. Weighted fractional coloring conjecture. One has

$$\chi^*(\mathcal{C},\vec h)\le (k-1)\max_{1\le i\le k}\chi^*(\mathcal{M}_i,\vec h).$$

The source states this as an equivalent formulation of the weighted matching-cover conjecture. It is proved for partition matroids, while the general case remains open.