Strong Fractional Tuza's conjecture

Let $G$ be a graph, and let $\nu(G)$ be the maximum number of edge-disjoint triangles in $G$. For $k\in\mathbb{N}$, a $k$-multi-transversal is a multiset $F\subseteq E(G)$ such that every triangle of $G$ contains at least $k$ elements of $F$. Define $\tau^*_k(G)$ to be the minimum of $|F|/k$ over all $k$-multi-transversals $F$.

Strong Fractional Tuza's conjecture. For every $k\in\mathbb{N}$ with $k\geq 2$ and every graph $G$,

$$\tau^*_k(G)\leq 2\nu(G).$$

This strengthens the fractional inequality $\tau^*(G)\leq 2\nu(G)$ by requiring a bounded-denominator multi-transversal for every $k\geq2$. The fractional inequality is known, but the stated uniform bound for all such $k$ is posed as an open question.