A Tight Fractional Version of Generalized Tuza's Conjecture

For an $r$-uniform hypergraph $H$, let $ν(H)$ be the maximum number of edges no two of which share $r-1$ vertices, and $τ(H)$ the minimum number of $(r-1)$-sets such that every edge contains one of them. Aharoni and Zerbib conjectured that $τ(H)\le\lceil\frac{r+1}{2}\rceil\,ν(H)$, which for $r=3$ ge