Aharoni–Zerbib's conjecture for generalized $(k-1)$-covers

For $m<k$, let $\mathcal H(k,m)$ be the family of hypergraphs obtained by taking all $m$-subsets of the edges of a $k$-uniform hypergraph, and let

$$h(k,m)=\sup\left\{\frac{\tau(J)}{\nu(J)}\mid J\in\mathcal H(k,m)\right\}.$$

Aharoni–Zerbib's conjecture for generalized $(k-1)$-covers.

$$h(k,k-1)=\left\lceil\frac{k+1}{2}\right\rceil.$$

This predicts the exact covering-to-matching ratio in the case of $(k-1)$-subsets. The source gives no resolution status.