Conjecture on the α-spectral radius of hypertrees in NC(m,d)

Let $NC(m,d)$ be the class of hypertrees with $m$ edges and diameter $d$, let $H_1(m,d)$ be the specified extremal hypertree, and let $\rho_{\alpha}$ denote the α-spectral radius. Assume $0\leq\alpha<1$.

The $NC(m,d)$ extremal conjecture. For $G\in NC(m,d)$,

$$\rho_{\alpha}(G)\leq\rho_{\alpha}(H_1(m,d)).$$

Equality holds if and only if $G\cong H_1(m,d)$. This conjecture would identify $H_1(m,d)$ uniquely as the maximizer of the α-spectral radius in $NC(m,d)$. The supplied text does not state whether it has been resolved.