Signless Laplacian supersaturation conjecture for cliques

Let $G$ be an $n$-vertex graph, and let $q(G)$ denote its signless Laplacian spectral radius. For fixed $k\geq 3$ and sufficiently large $n$, consider graphs whose signless Laplacian spectral radius exceeds the threshold $2(1-1/k)n$. Signless Laplacian supersaturation conjecture. If $k\geq 3$ is fixed and $n$ is sufficiently large, then

$$q(G)>\left(1-\frac{1}{k}\right)2n$$

for an $n$-vertex graph $G$ implies that $G$ contains at least

$$\left(\frac{n}{k}\right)^{k-1}+O(n^{k-2})$$

copies of $K_{k+1}$. This would provide a signless-Laplacian analogue of clique supersaturation, while the paper notes that the corresponding assertion for triangles fails and that the conjecture differs from the stronger linear-margin supersaturation theorem.