The spectral extremal conjecture for 3-chromatic graph families

Let $\mathcal F$ be a finite graph family, let $\chi(F)$ denote the chromatic number of a graph $F$, let $\mathrm{ex}(n,\mathcal F)$ be the maximum number of edges in an $\mathcal F$-free graph on $n$ vertices, let $T_{n,2}$ be the Turán graph with two parts, and let $\mathrm{SPEX}(n,\mathcal F)$ and $\mathrm{EX}(n,\mathcal F)$ denote, respectively, the spectrally extremal and extremal $\mathcal F$-free graphs. For sufficiently large $n$, suppose that

$$\chi(F)=3,\qquad \mathrm{ex}(n,\mathcal F)\le e(T_{n,2})+Qn,$$

where $Q<\frac{1}{3}$. The spectral extremal conjecture. For $n$ large enough,

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

This is the proposed weaker analogue, for $r=2$, of the paper's spectral extremal result for non-bipartite graph families with $r\geq 3$. The preceding proposition shows that the conclusion fails at the boundary value $Q=\frac{1}{3}$ for an infinite sequence of $n$; the conjectured range below that threshold remains open.