Asymptotic extremal conjecture for 3-uniform hypergraphs with bounded matching number

Let $F$ be a graph, let $\rchi(F)$ denote its chromatic number, let $q(F)$ be the minimum number of red vertices in a strong red-blue coloring of $F$, and let $H_i$ be the 3-uniform hypergraphs constructed from the ordered link graphs of the red vertices of $F$. For positive integers $s$ and sufficiently large $n$, consider the extremal number forbidding $F$ and the matching $M^3_{s+1}$. The asymptotic extremal conjecture. If $\rchi(F)=2$ and $q(F)\le s$, then

$$\operatorname{ex}_3(n,\{F,M^3_{s+1}\})=\max\{|E(H_i)|: i\le q(F)\}+o(n^2).$$

This conjecture proposes that the constructions $H_i$ give the asymptotically sharp lower bound for the corresponding hypergraph Turán problem; the source provides the construction and its $F$-freeness, but no resolution of the conjecture.