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Let $\mathcal T (n, k)$ be the set of the $k$-uniform supertrees with $n$vertices and $m$ edges, where $k\geq 3$, $n\geq 5$ and $m=\frac{n-1}{k-1}$. %Let $m$ be the number of the edges of the supertrees in $\mathcal T (n, k)$,where $m=\frac{n-1}{k-1}$. A conjecture concerning the supertrees with thefourth through the eighth largest $\alpha$-spectral radii in $\mathcal T (n,k)$ was proposed by You et al.\ (2020), where $0 \leq \alpha<1$, $k\geq 3$ and$m \geq 10$. This conjecture was partially solved for $1-\frac{1}{m-2}\leq\alpha <1$ and $m\geq 10$ by Wang et al.\ (2022). When $0\leq \alpha<1-\frac{1}{m-2}$ and $m \geq 10$, whether this conjecture is correct or notremains a problem to be further solved. By using a new $\rho_{\alpha}$-normallabeling method proposed in this article for computing the $\alpha$-spectralradius of the $k$-uniform hypergraphs, we completely prove that this conjectureis right for $0\leq\alpha<1$ and $m\geq 13$.

A complete solution of the $k$-uniform supertrees with the eight largest $\alpha$-spectral radii