Edge-localized Turán inequality for the signless Laplacian spectral radius

Let $G$) be a connected graph, let $E(G)$ be its edge set, let $\deg(u)$ denote the degree of a vertex $u$, let $q(G)$ denote the spectral radius of the signless Laplacian of $G$, and let $c(uv)$ denote the relevant clique parameter associated with the edge $uv$. Edge-localized Turán conjecture. For any connected graph $G$,

$$q(G)\le 2\sum_{uv\in E(G)}\left(1-\frac{1}{c(uv)}\right)\left(\frac{1}{\deg(u)} + \frac{1}{\deg(v)}\right).$$

This is proposed as an edge-localized analogue of a Turán-type inequality for the signless Laplacian. The supplied text does not state whether the conjecture has been resolved.