= Solution
Because $T$ itself is a tree, every finite induced connected exhaustion has the unique spanning tree consisting of all its edges. Thus its free spanning forest is deterministically $T$.
By the stated transience criterion, choose an edge $e=uv$ whose two complementary subtrees are transient. Run <Wilson algorithm rooted at infinity> first from $u$. With positive probability its loop-erased walk remains forever in the $u$-side. Starting next from $v$, there is likewise positive conditional probability that its walk remains forever in the $v$-side. On this event the two rays never use $e$, so $e$ is absent from the <wired uniform spanning forest>. The wired law is therefore not the deterministic free law.
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