= Solution
Project $X$ to its label in $\mathbb Z^2$. At each step the label makes a nearest-neighbor move with probability $4/5$ and holds with probability $1/5$ when $X$ crosses between layers. This lazy planar random walk is recurrent because ordinary random walk on $\mathbb Z^2$ is recurrent.
Whenever the label returns to that of $x_0$, the layer coordinate belongs to a two-state irreducible chain, and there is a uniformly positive chance to equal the original layer. Infinitely many label returns therefore give infinitely many returns to $x_0$. Thus simple random walk on $G$ is recurrent.
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