 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.