Bollobás–Nikiforov signless Laplacian conjecture for clique-free graphs

Let $G$ be a graph containing no copy of $K_{r+1}$, and let $m$ be its number of edges. Write $q(G)$ for the signless Laplacian spectral radius. Bollobás–Nikiforov conjecture. One has

$$q(G)\le \sqrt{8m\left(1-\frac{1}{r}\right)}.$$

This is the signless Laplacian analogue of the corresponding adjacency spectral-radius bound for $K_{r+1}$-free graphs; the source gives no resolution of the conjecture.