Conjecture that color-critical forbidden graphs inherit weak Turán-goodness

Let $F$ be a graph with chromatic number $r+1$ and a color-critical edge, meaning an edge whose removal decreases the chromatic number. Let $H$ be weakly $F$-Turán-good if, for all sufficiently large $n$, ${\mathrm{ex}}(n,H,F)={\mathcal N}(H,T)$ for some complete $r$-partite $n$-vertex graph $T$. Color-critical inheritance conjecture. If $H$ is weakly $K_{r+1}$-Turán-good, then $H$ is weakly $F$-Turán-good. This proposes that color-critical forbidden graphs behave like cliques in this generalized Turán problem; the statement is presented as an expectation and remains open in the source.