Cioabă, Desai and Tait's spectral Turán containment conjecture

Let $F$ be a graph. Write $\operatorname{EX}(n,F)$ for the family of $F$-free graphs attaining the extremal number, and let $\operatorname{SPEX}(n,F)$ denote the family of $F$-free graphs attaining the maximum spectral radius. A graph family is Turán graphs plus $O(1)$ edges if each of its graphs can be obtained from the appropriate Turán graph by adding $O(1)$ edges. Cioabă, Desai and Tait's conjecture. If the graphs in $\operatorname{EX}(n,F)$ are Turán graphs plus $O(1)$ edges, then, for sufficiently large $n$,

$$\operatorname{SPEX}(n,F)\subseteq\operatorname{EX}(n,F).$$

This proposes a general condition under which spectral extremal graphs are also extremal in the ordinary Turán problem; the source gives examples where the containment is known, but does not state a resolution of the conjecture.