The hereditary-family polychromatic coloring conjecture

Let $\mathcal F$ be a hereditary family of hypergraphs. For $k\ge1$, let $m_k(\mathcal F)$ be the smallest integer $m$ such that every $m$-heavy hypergraph in $\mathcal F$ has a polychromatic $k$-coloring, with $m_k(\mathcal F)=\infty$ if no such integer exists. Hereditary-family polychromatic coloring conjecture. If

$$m_2(\mathcal F)<\infty,$$

then

$$m_k(\mathcal F)<\infty$$

for every $k\ge2$. The conjecture is known when $m_2(\mathcal F)=2$, in which case Berge proved $m_k(\mathcal F)=k$ for every $k$; it is open in general, and the hereditary hypothesis is necessary.