Fractional covering conjecture for looms

Let $A$ and $B$ be hypergraphs forming an $(r,s)$-loom, and write $\mathbb{L}=(A,B)$. Define the fractional covering number of the loom by

$$\tau^*(\mathbb{L})=\tau^*(A\cup B).$$

Here $\tau^*$ is the minimum total weight of a nonnegative fractional vertex cover.

Loom fractional covering conjecture. If $\mathbb{L}=(A,B)$ is an $(r,s)$-loom, then

$$\tau^*(\mathbb{L})=\max(r,s).$$

The source has the upper bound $\tau^*(\mathbb{L})\leqslant\max(r,s)$ and proposes equality; it also notes that this conjecture would imply the Gyárfás–Lehel conjecture in the equal-uniformity setting. It remains open.