Recurrence conjecture for the one-dimensional centre of mass

Let $X$ be a one-dimensional random-walk increment with zero mean, and let $G_n$ denote the centre of mass at time $n$. Recurrence conjecture. Suppose $d=1$. If

$$\operatorname{\mathbb{E}}X=0,$$

then $G_n$ is recurrent. This is proposed as the one-dimensional zero-drift case of the centre-of-mass problem; no resolution is given in the source.