You and Yang's signless Laplacian power-sum conjecture for bipartite graphs

Let $G$ be a bipartite graph with $n$ vertices. For a graph $G$, let $q_1(G),\ldots,q_r(G)$ be its nonzero signless Laplacian eigenvalues, and define

$$S_\alpha(G):=q_1(G)^\alpha+\cdots+q_r(G)^\alpha.$$

You and Yang's conjecture. If $\alpha>1$, then

$$S_\alpha(G)\leq n^\alpha+\left(\lfloor n/2\rfloor-1\right)\lceil n/2\rceil^\alpha+(\lceil n/2\rceil-1)\lfloor n/2\rfloor^\alpha,$$

with equality if and only if $G=K_{\lfloor n/2\rfloor,\lceil n/2\rceil}$.

This conjecture complements the known inequality for connected bipartite graphs when $0<\alpha\leq1$ and predicts that the balanced complete bipartite graph maximizes the signless Laplacian power sum for powers greater than one. Its validity is stated as open in the source.