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Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in  that the Free Uniform Spanning Forest (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 FSF exhibits a phase transition. For large enough w we show that the FSF is connected, while for adoi:10.1214/22-ecp453 fatcat:kxidgiyb7nhefgte3gwh4ieb3m