Gyárfás–Lehel covering conjecture for cross-intersecting partite hypergraphs

Let $A$ and $B$ be non-empty cross-intersecting hypergraphs: every edge of $A$ meets every edge of $B$. Suppose both are $r$-partite and share the same $r$-partition. A cover of $A\cup B$ is a set of vertices meeting every edge, and $\tau(A\cup B)$ denotes its minimum size.

Gyárfás–Lehel conjecture. Then

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

The bound improves the elementary bound obtained from the union of one edge of each hypergraph. The source presents this as open and notes that the bound is tight if true.