Li, Feng, and Liu's signless Laplacian spectral Turán conjecture

Let $G$ be a graph with $m$ edges. A graph is $K_{r+1}$-free if it contains no complete subgraph on $r+1$ vertices. Let $q(G)$ denote the largest eigenvalue of the signless Laplacian matrix of $G$. Li, Feng, and Liu's conjecture. If $G$ is $K_{r+1}$-free, then

$$q(G) \leq \sqrt{8\Big( 1-\frac{1}{r} \Big) m}.$$

The conjecture proposes a signless-Laplacian analogue of a spectral Turán inequality. The source presents it as a conjecture from Li, Feng, and Liu; no resolution is given here.