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 $α$-spectral radii in $\mathcal T (n, k)$ was proposed by You et al.\ (2020), where $0 \leq α<1$, $k\geq 3$ and $m \geq 10$. This conjecture was partially solved for $1-\frac{1}{m-2}\leq α<1$ and $m\geq 10$ by Wang et al.\ (2022). When $0\leq α<1-\frac{1}{m-2}$ and $m \geq 10$, whether this conjecture is correct or not remains a problem to be further solved. By using a new $ρ_α$-normal labeling method proposed in this article for computing the $α$-spectral radius of the $k$-uniform hypergraphs, we completely prove that this conjecture is right for $0\leqα<1$ and $m\geq 13$.