Star-minimization conjecture for the least $A_\alpha$-eigenvalue

Let $G$ be a connected graph on $n$ vertices with $\frac{1}{2}<\alpha<1$. Star-minimization conjecture.

$$\lambda_n(A_{\alpha}(G))\geq\lambda_n(A_{\alpha}(K_{1,n-1})),$$

with equality if and only if $G\cong K_{1,n-1}$. This conjecture asks whether the star minimizes the least eigenvalue of $A_\alpha$ among connected graphs of order $n$ in the specified range of $\alpha$; no resolution is stated in the supplied text.