Smith normal form equivalence for rooted-tree products

Let $\mathsf{T}$ be the tree in Fig., and let $T$ be any nontrivial tree with root $r$. Define $G_1=T(r) \odot \mathsf{T}(4)$ and $G_2=T(r) \odot \mathsf{T}(7)$. Smith normal form conjecture. The matrices $tI-L_\mu(G_1)$ and $tI-L_\mu(G_2)$ should have the same Smith normal form over $\mathbb{Q}(\mu)[t]$. The claim extends the preceding proved result from paths and stars to arbitrary nontrivial rooted trees; its status is not resolved in the supplied text.