Polychromatic finiteness conjecture for hereditary hypergraph families

Let $\mathcal F$ be a hereditary family of hypergraphs, and let $m_k(\mathcal F)$ be the least integer $m$, if it exists, such that every $m$-heavy member of $\mathcal F$ is polychromatically $k$-colorable; set $m_k(\mathcal F)=\infty$ if no such $m$ exists. A hypergraph is $m$-heavy when every edge has size at least $m$, and a coloring is polychromatic when every edge contains a vertex of each of the $k$ colors.

Polychromatic finiteness conjecture. If $m_2(\mathcal F)<\infty$, then $m_k(\mathcal F)<\infty$ for every $k$ for any hereditary family $\mathcal F$.

This conjecture is known when $m_2(\mathcal F)=2$, by a result of Berge, but remains open in general. It asks whether finiteness of the threshold for 2-colorability forces finiteness of the corresponding threshold for every number of colors.