Gerbner's asymptotic conjecture for forbidden bipartite 3-graphs with bounded matching number

Let $F$ be a $3$-graph, let $\chi(F)$ denote its chromatic number, and let $q(F)$ be the parameter defined by the ordered link-graph chromatic numbers in the source. For $i\in[q(F)]$, let $H_i$ be the $F$-free $3$-graphs constructed there, with matching number at most $s$. Assume that $\chi(F)=2$, $q(F)\leq s$, and $n$ is sufficiently large. Gerbner's asymptotic conjecture.

$$\mathrm{ex}\bigl(n,\{F,M_{s+1}^3\}\bigr)=\max\{\lvert E(H_i)\rvert:i\in[q(F)]\}+o(n^2).$$

The constructions $H_i$ provide competing $F$-free examples with matching number at most $s$, motivating the conjectured asymptotic extremal value. The source gives no resolution evidence for this conjecture.