The linear-growth conjecture for hereditary-family polychromatic thresholds

Let $\mathcal F$ be a hereditary family of hypergraphs, and let $m_k(\mathcal F)$ be the least threshold guaranteeing a polychromatic $k$-coloring for every $m$-heavy member of $\mathcal F$. Linear-growth conjecture. If $m_2(\mathcal F)<\infty$, then

$$m_k(\mathcal F)=O(k)$$

for every $k$. This is a stronger version of the finiteness conjecture above; the source gives no general proof or disproof.