Positive exponential percolation dimension for super-polynomial growth groups

Let $G$ be a finitely generated group of super-polynomial growth, and let ${}^{e}\operatorname{pdim}(G)$ denote its exponential percolation dimension. Positive-dimension conjecture. For every finitely generated group $G$ of super-polynomial growth,

$${}^{e}\operatorname{pdim}(G)>0.$$

This is presented as a plausible strengthening related to the Gap Conjecture: it predicts that every finitely generated group whose growth is not polynomial has positive exponential percolation dimension. No resolution is given in the source.