Cutset finiteness conjecture for graphs of dimension greater than one

Let $G$ be a graph, and let $\operatorname{Dim}(G)$ denote its dimension parameter. Cutset finiteness conjecture. Every graph satisfying

$$\operatorname{Dim}(G)>1$$

has

$$\kappa<\infty.$$

The conjecture proposes that finiteness of the cutset-growth parameter is exactly the geometric criterion corresponding to $p_c<1$. The source explicitly states that this remains an important open problem; in particular, it is not resolved by the paper's results.