Pinnable covering conjecture for hypergraphs of different uniformities

Let $A$ be an $r$-uniform hypergraph and $B$ an $s$-uniform hypergraph on the same vertex set, so $V(A)=V(B)$. Assume that $A$ and $B$ are cross-intersecting. A hypergraph is pinnable if it has a set meeting every edge in exactly one vertex.

General pinnable covering conjecture. If $A\cup B$ is pinnable, then

$$\tau(A\cup B)\leqslant r+s-2.$$

The source proves this when $s=2$, but leaves the general case open.