 Solution

Fix a simple path of 2022 vertices in the first layer. The event that all its edges belong to the uniform spanning tree has positive probability: a finite acyclic edge set can be extended to a spanning tree in every sufficiently large finite exhaustion, and the transfer-current determinant for that forest has a positive infinite-volume limit.

The event that $T_1$ has a component of size at least 2022 is invariant under translations of ${\mathbb{Z}}^2$. The uniform spanning-tree law on this transitive recurrent graph is translation ergodic, so an invariant event of positive probability has probability one. Therefore $T_1$ almost surely has such a component.