  Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected
transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform
Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the
dependence of this connectedness on $H$ remains somewhat mysterious. We study
the case when a positive weight $w$ is put on the edges of the $H$-copies in
$G$, and conjecture that the connectedness of the $\mathsf{FSF}$ exhibits a
phase transition. For large enough $w$ we show that the $\mathsf{FSF}$ is
connected, while for a large family of $H$ and $T$, the $\mathsf{FSF}$ is
disconnected when $w$ is small (relying on arXiv:2006.06387). Finally, we prove
that when $H$ is the graph of one edge, then for any $w$, the $\mathsf{FSF}$ is
a single tree, and we give an explicit formula for the distribution of the
distance between two points within the tree.
