Signless Laplacian spectral extremal graph conjecture for graph families

Let $\mathcal{F}$ be a finite family of graphs, let $\chi(\mathcal{F})$ denote its chromatic number, and let $\operatorname{ex}(n,\mathcal{F})$ be the maximum number of edges in an $n$-vertex $\mathcal{F}$-free graph. Let $T_{n,k}$ be the Turán graph and let $\operatorname{Ex}(n,\mathcal{F})$ be the set of $n$-vertex $\mathcal{F}$-free graphs with the maximum number of edges. Signless Laplacian spectral extremal graph conjecture. If $\mathcal{F}$ is a finite graph family with $\chi(\mathcal{F})=k+1\geq 4$ and

$$\operatorname{ex}(n,\mathcal{F})<e(T_{n,k})+\left\lfloor\frac{n}{2k}\right\rfloor,$$

then, for sufficiently large $n$, every $n$-vertex graph $G$ achieving the maximal signless Laplacian spectral radius among all $n$-vertex $\mathcal{F}$-free graphs belongs to $\operatorname{Ex}(n,\mathcal{F})$. This proposes that under the stated near-Turán extremal condition, signless-Laplacian spectral extremality and edge extremality have the same graphs; the cited adjacency-spectral analogue is known, but the signless-Laplacian version remains open in the supplied text.