Lin–Miao–Guo conjecture on the maximum $A_{\alpha}$-spread

Let $G$ be a connected simple undirected graph with $n\geq 5$ vertices. Let $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $D(G)$ is the diagonal degree matrix, $A(G)$ is the adjacency matrix, and $\alpha\in[0,1]$. If $\lambda_1^{(\alpha)}(G)$ and $\lambda_n^{(\alpha)}(G)$ are respectively the largest and smallest eigenvalues of $A_{\alpha}(G)$, define the $A_{\alpha}$-spread by

$$S_{\alpha}(G)=\lambda_1^{(\alpha)}(G)-\lambda_n^{(\alpha)}(G).$$

Lin–Miao–Guo conjecture. If $1/2\leq\alpha<1$, then

$$S_{\alpha}(G)\leq S_{\alpha}(Ki_{n,n-1}),$$

with equality if and only if $G\cong Ki_{n,n-1}$.

This conjecture proposes the extremal connected graph for the $A_{\alpha}$-spread when $\alpha$ is at least one half. The supplied text does not state whether it has been resolved; the notation $Ki_{n,n-1}$ is used in the source but is not defined in the provided context.