= Solution
Run <Wilson algorithm rooted at infinity> with $w_1,\ldots,w_K$ first in the enumeration. Couple its first $K$ walks with the independent walks in the hypothesis. On the positive-probability event that their ranges are pairwise disjoint, no walk from $w_j$ hits any earlier loop-erased range. Wilson's algorithm therefore creates $K$ distinct trees, so
$$
\mathbb P(\text{the wired uniform spanning forest has at least $K$ trees})>0.
$$
The <component-number zero-one law for the wired uniform spanning forest> says that its number of trees is almost surely constant; it follows from tail triviality of the wired uniform spanning forest and the fact that all its trees are infinite. The displayed positive probability must consequently equal one.
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