Structural conjecture for near-extremal $K_{r+1}$-free graphs

Let $r\geq 2$ be an integer, let $n$ be sufficiently large, and let $D_r(G)$ denote the minimum number of edges that must be deleted from a graph $G$ to make it $r$-partite. For graphs $G$ and $H$, write $G\otimes H$ for their join, and let an unbalanced blow-up mean a blow-up whose vertex classes need not have equal sizes.

Structural conjecture. There exists $\alpha_0>0$ such that, for every $0\leq\alpha\leq\alpha_0$ and every $K_{r+1}$-free graph $G$ on $n$ vertices, there is an unbalanced blow-up $H$ of $K_{r-2}\otimes C_5$ on $n$ vertices satisfying

$$e(H)\geq e(G)$$

and

$$D_r(G)\leq D_r(H).$$

The conjecture proposes that the worst examples for the distance from $r$-partiteness among near-extremal $K_{r+1}$-free graphs have the form of unbalanced $K_{r-2}\otimes C_5$ blow-ups. The supplied text does not state whether this structural statement has been resolved.