Conjecture on the strong chromatic extremal function of Berge paths

Let $r\geq 3$ be a fixed integer. For a path $P_k$ with $k$ edges, let $\mathcal{B}(P_k)$ denote its Berge hypergraph, and let $\mathrm{schex}_r(n,\mathcal{B}(P_k))$ be the strong chromatic extremal function for $r$-uniform hypergraphs avoiding $\mathcal{B}(P_k)$. Path strong-coloring conjecture. There exists a function $f(r)$ such that, whenever $k\geq f(r)$, we have

$$\mathrm{schex}_r(n,\mathcal{B}(P_k))=k.$$

The case $r=3$ is established in the preceding theorem, while the conjecture predicts the same equality for every fixed uniformity $r\geq 3$ once the path length is sufficiently large.