Zhu's conjecture for categorical products of hypergraphs

Let $\mathcal{G}=(V,E)$ and $\mathcal{G}'=(V',E')$ be hypergraphs. Their categorical product $\mathcal{G}\times\mathcal{G}'$ has vertex set $V\times V'$ and edges whose projections onto the two factors are edges of the respective hypergraphs. Let $\chi(\mathcal{G})$ denote the chromatic number, the least number of colors in a proper coloring with no monochromatic edge. Zhu's conjecture. The categorical product of two hypergraphs satisfies

$$\chi(\mathcal{G}\times\mathcal{G}')=\min\{\chi(\mathcal{G}),\chi(\mathcal{G}')\}.$$

The conjecture is presented as the hypergraph generalization of Hedetniemi's conjecture. The paper states that it proves the conjecture for usual Kneser hypergraphs of the same rank and obtains a more general lower bound for products of Kneser hypergraphs of common rank; the unrestricted hypergraph statement is not claimed to be resolved here.