Chen–Liu–Zhang signless Laplacian conjecture for intersecting cliques

For integers $k\ge 1$ and $r\ge 3$, let $F_{k,r}$ be the forbidden graph from the source, let $S_{n,a}=K_a\vee I_{n-a}$, and let $q(G)$ denote the signless Laplacian spectral radius. Chen–Liu–Zhang conjecture. There exists an integer $n_0(k,r)$ such that, if $n\ge n_0(k,r)$ and $G$ is an $F_{k,r}$-free graph on $n$ vertices, then

$$q(G)\le q(S_{n,k(r-2)}),$$

with equality if and only if $G=S_{n,k(r-2)}$. The case $r=3$ is identified in the source with a theorem of Zhao, Huang and Guo; the general assertion remains open.