Path-maximizes analytic spread among trees

Let $T_1$ and $T_2$ be trees on $n$ and $m$ vertices, respectively, and fix $i\in V(T_1)$ and $j\in V(T_2)$. For a positive integer $l$, define $G_1$ as the tree on $n+m+l$ vertices with edge set

$$E(T_1)\sqcup E(T_2)\sqcup\\{\\{i,m+n+1\\},\\{m+n+1,m+n+2\\},\ldots,\\{m+n+l,j\\}\\},$$

and define $G_2$ as the tree on $n+m+l$ vertices obtained by identifying $i$ with $j$ in the disjoint union of $T_1$ and $T_2$, then adjoining the tail with edges $\\{i,n+m+1\\},\\{n+m+1,n+m+2\\},\ldots,\\{m+n+l,m+n+l+1\\}$. Path-maximization conjecture. For every graph $H$,

$$\ell(J_{G_1,H})\geq\ell(J_{G_2,H}).$$

The proposed comparison says that separating the two trees and joining them by a path does not decrease analytic spread relative to identifying the attachment vertices and adding a tail. The source notes that this would imply $\ell(J_{P_n,H})\geq\ell(J_{T,H})$ for every $n$-vertex tree $T$ and every graph $H$, but gives no proof or resolution.