Aharoni–Zerbib's extremal ratio conjecture for generalized matchings

For $m<k$, let $\mathcal H(k,m)$ be the family of hypergraphs $H^{(m)}$ arising from $k$-uniform hypergraphs, and define

$$g(k,m)=\max\{\tau(J)\mid J\in\mathcal H(k,m),\ \nu(J)=1\},\qquad h(k,m)=\sup\left\{\frac{\tau(J)}{\nu(J)}\mid J\in\mathcal H(k,m)\right\}.$$

Here $\nu$ and $\tau$ denote matching and covering numbers.

Aharoni–Zerbib's extremal ratio conjecture.

$$h(k,m)=g(k,m)\quad\text{for all }m\leq k.$$

If true, the largest covering-to-matching ratio in this class would always be attained by a hypergraph with matching number one. The source gives no resolution status.