Keszegh–Pálvölgyi linear bound conjecture for polychromatic coloring

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.

Keszegh–Pálvölgyi linear bound conjecture. For every $k$ and every hereditary family $\mathcal F$,

$$m_k(\mathcal F)\le (k-1)(m_2(\mathcal F)-1)+1.$$

This is a stronger quantitative form of the finiteness conjecture and was conjectured by Keszegh and the author. The paper states that it is disproved by a 5-uniform hypergraph with no polychromatic 3-coloring, although all its restricted subhypergraphs with edges of size at least 3 are 2-colorable.