Li–Feng pendent-path transfer conjecture for the alpha-index

Let $G$ be a connected graph, let $u$ be a vertex of $G$, and let $G_{p,q}(u)$ denote the graph obtained by attaching pendent paths of lengths $p$ and $q$ at $u$. For $\rho_{\alpha}$, the alpha-index of a graph, let $\alpha\in[0,1)$. Pendent-path transfer conjecture. If $p\geq q+2\geq3$, then

$$\rho_{\alpha}\left(G_{p,q}(u)\right)<\rho_{\alpha}\left(G_{p-1,q+1}(u)\right).$$

The assertion is known for $\alpha=0$ and $\alpha=1/2$, and the paper proves it for all $\alpha\in[0,1)$ when $\rho_{\alpha}(G_{p,q}(u))\geq9/4$. Its validity without that additional spectral-radius condition remains open.