Pinnable cross-intersection covering conjecture

Let $A$ and $B$ be cross-intersecting $r$-uniform hypergraphs. A hypergraph $H$ is pinnable if there is a set $p$ such that $|p\cap e|=1$ for every edge $e\in H$; equivalently, $H^\perp\neq\varnothing$.

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

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

This weakens the common-partition hypothesis in the Gyárfás–Lehel conjecture. The source presents it as a conjecture and does not establish it in general.