Gerbner–Tompkins–Zhou's asymptotic conjecture for expanded graphs with bounded matching number

Let $G$ be a graph with chromatic number $\chi(G)=\ell>r$, and let $U$ be an independent set of $G$ such that deleting $U$ results in a graph $G_0$ of chromatic number $\ell-1$. Suppose two color classes of $G_0$ have $m'(G)$ edges between them, and write $m(G')$ for the quantity appearing in the conjectured asymptotic formula. Gerbner–Tompkins–Zhou's conjecture. If $s$ is sufficiently large, then

$${\mathrm{ex}}_r(n,\{G^{(r)+},M_{s+1}^{r+}\})=(m(G')-1)\binom{n}{r-1}+(s-m'(G)+1)\binom{\ell-2}{r-1}\left(\frac{n}{\ell-2}\right)^{r-1}+o(n^{r-1}).$$

This proposes the asymptotic Turán number for expansions of graphs with bounded matching number, extending the preceding hypergraph results; the supplied text gives no resolution.