= Solution
Fix a finite nonempty vertex set $K$. Once $G_n$ contains $K$ and every edge incident to it, every spanning tree of $G_n$ contains an edge of the finite cut
$$
\partial_EK=\{uv:u\in K,\ v\notin K\},
$$
because otherwise $K$ is disconnected from the rest of $G_n$. Hence
$$
\mu_n(\text{every edge of }\partial_EK\text{ is absent})=0,
$$
and the same holds in the weak limit. If the free spanning forest had a finite component, its vertex set would be some finite connected $K$ and all edges of $\partial_EK$ would be absent. Taking the countable union over finite $K$ proves that every component is infinite almost surely.
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