Yuster's hypergraph independent transversal asymptotic conjecture

For integers $n,k,r,s$, let $f(k,r,s)$ be the extremal function for independent transversals in $(n,k,r,s)$-graphs. Fix $r\ge 3$. Yuster's conjecture. For all $1\le s\le k$,

$$f(k,r,s)=(r-1+o_k(1))\sqrt[r-1]{\frac{k^r}{s}}.$$

The paper proves the asserted order of magnitude when $s\ge (r-1)!$, and proves the corresponding upper bound for $s\ge C\log k$, but the full range, especially small $s$, remains open.