Universality conjecture for simple random walk on growing domains

Let $d\ge3$, and let $\bmathbb{D}_t\subseteq\mathbb{Z}^d$ be connected and non-decreasing, with $f(t)\mathbf{B}_1\cap\mathbb{Z}^d\subseteq\mathbb{D}_t\subseteq f(t)\mathbf{B}_c\cap\mathbb{Z}^d$ for a finite $c$ and a non-decreasing, unbounded, strictly positive function $f(t)$. Let $\bmathbb{Y}_t$ be simple random walk on $\bmathbb{D}_t$, starting at $Y_0=0$, and set $J_f:=\int_0^\infty f(t)^{-d}\,dt$. Universality conjecture. Almost surely, $\bmathbb{Y}_t$ returns to the origin finitely often if and only if

$$J_f<\infty.$$

This conjecture asserts that only the asymptotic growth rate of the domains matters for recurrence versus transience. The paper proposes it and proves partial results for simple random walk on subgraphs of $\bmathbb{Z}^d$.