Signless Laplacian power-sum conjecture for bipartite graphs

Let $G$ be a bipartite graph with $n$ vertices. Write $Q(G)$ for its signless Laplacian matrix, let $q_1,\dots,q_n$ be the eigenvalues of $Q(G)$, and define

$$S_\alpha(G)=\sum_{i=1}^n q_i^\alpha.$$

Bipartite signless Laplacian power-sum conjecture. If $\alpha>1$, then

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

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

This extends the established extremal result for the corresponding range of exponents in related graph classes to bipartite graphs. The conjecture concerns the sharp upper bound and its equality case for all $\alpha>1$.