Asymptotic extremal conjecture for powers of graphs with bounded matching number

Let $G$ be a graph with chromatic number $k$, let $G^r$ denote its $r$-uniform power, and let $m'(G)$ be the number of edges between a specified pair of color classes of a $(k-1)$-coloring of the graph obtained after deleting an independent set $U$. Assume that deleting $U$ yields a graph $G_0$ of chromatic number $k-1$. Asymptotic extremal conjecture. If $k>r$, there is an independent set $U$ of $G$ such that $G_0=G-U$ has chromatic number $k-1$, two color classes of $G_0$ have $m'(G)$ edges between them, and $s$ is sufficiently large, then

$$\operatorname{ex}(n,\{G^r,M^r_{s+1}\})=(m'(G)-1)\binom{n}{r-1}+(s-m'(G)+1)\binom{k-2}{r-1}\left(\frac{n}{k-2}\right)^{r-1}+o(n^{r-1}).$$

The conjecture asserts that the construction described in the source gives the asymptotically correct extremal value when the matching bound is sufficiently large; the source presents this as a belief and does not provide a resolution.