Pendent-path transfer conjecture for adjacent attachment vertices

Let $G$ be a connected graph, and let $u$ and $v$ be adjacent vertices of $G$ with degree at least $2$. Let $G_{p,q}(u,v)$ denote the graph obtained by attaching pendent paths of lengths $p$ and $q$ at $u$ and $v$, respectively, and let $\rho_{\alpha}$ be the alpha-index. Pendent-path transfer conjecture. If $\alpha\in[0,1)$, $q\geq1$, and $p\geq q+2$, then

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

This is posed as a related open problem after the first conjecture. The source gives no resolution for this adjacent-vertex version.