You and Yang's connectivity-constrained signless Laplacian power-sum conjecture

Let $G$ be a graph with $n$ vertices and vertex connectivity $\kappa(G)\leq k$. Let $q_1(G),\ldots,q_r(G)$ be the nonzero signless Laplacian eigenvalues of $G$, and define

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

Set

$$b_\alpha(n,k)=k(n-2)^\alpha+(n-k-2)(n-3)^\alpha+\left(n-2+\frac{k}{2}+\frac{1}{2}\sqrt{(k-2n)^2+16(k-n+1)}\right)^\alpha$$ $$\qquad+\left(n-2+\frac{k}{2}-\frac{1}{2}\sqrt{(k-2n)^2+16(k-n+1)}\right)^\alpha.$$

You and Yang's conjecture. (i) If $0<\alpha<1$, then

$$S_\alpha(G)\leq b_\alpha(n,k),$$

with equality if and only if $G=K_k\vee(K_1\cup K_{n-k-1})$. (ii) If $G$ is connected and $\alpha<0$, then

$$S_\alpha(G)\geq b_\alpha(n,k),$$

with equality if and only if $G=K_k\vee(K_1\cup K_{n-k-1})$.

These assertions extend the preceding connectivity bound, which was established for $\alpha\geq1$, and address the unsettled ranges $0<\alpha<1$ and $\alpha<0$. The source states that the validity of the conjecture for $-1\leq\alpha\leq1$ remains open.