Gerbner's color-critical-edge conjecture for weak Turán-goodness

A graph $H$ is weakly $K_k$-Turán-good when, for all sufficiently large $n$, some complete $(k-1)$-partite graph attains ${\mathrm{ex}}(n,H,K_k)$. An edge of a graph $F$ is color-critical if deleting it lowers the chromatic number. The Gerbner conjecture. If $H$ is weakly $K_k$-Turán-good, then $H$ is weakly $F$-Turán-good for every $k$-chromatic graph $F$ with a color-critical edge. The source states this as a conjecture in its discussion of forbidden graphs that are not cliques; its resolution status is not given.