  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 the
fourth 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 not
remains a problem to be further solved. By using a new $\rho_{\alpha}$-normal
labeling method proposed in this article for computing the $\alpha$-spectral
radius of the $k$-uniform hypergraphs, we completely prove that this conjecture
is right for $0\leq\alpha<1$ and $m\geq 13$.
