The gap conjecture for exponential percolation dimension

Let $G$ be a finitely generated group, and let ${}^{e}\operatorname{pdim}(G)$ denote its exponential percolation dimension. Gap conjecture. There is a $\beta>0$ such that

$${}^{e}\operatorname{pdim}(G)<\beta \implies {}^{e}\operatorname{pdim}(G)=0$$

for every finitely generated group $G$. This is proposed as an analogue for exponential percolation dimension of the geometric-group-theoretic Gap Conjecture, which asserts that sufficiently subexponential growth forces polynomial growth; the source presents this as an open problem.