Weighted covering conjecture for d-interval hypergraphs

Let $H$ be a hypergraph of $d$-intervals, let $\tau_w(H)$ be its weighted covering number, and let $\nu_w(H)$ be its weighted matching number. Weighted covering conjecture. One should have

$$\tau_w(H) \le d^2\nu_w(H).$$

This bound would improve the proved general estimate $\tau_w(H)\le 2d^2\nu_w(H)$ and would follow in the separated case from the weighted fractional covering conjecture. Its validity in the general case remains open.