The first-return asymptotic conjecture for transient transitive graphs

Let $G=(V,E)$ be a locally finite, connected, transitive, transient graph. Let $u_n$ be the $n$-step return probability, $f_n$ the first-return probability at time $n$, $\rho$ the spectral radius, and $F(z)=\sum_{n=1}^{\infty}f_nz^n$. Let $\mathsf{d}$ be the period of simple random walk on $G$,

$$\mathsf{d}:=\gcd\{n\geq 1\colon u_n>0\}\in\{1,2\}.$$

The first-return asymptotic conjecture. One has $f_n\asymp u_n$, and more precisely

$$\lim_{n\to\infty,\,\mathsf{d}\mid n}\frac{f_n}{u_n}=\bigl(1-F(\rho^{-1})\bigr)^2\in(0,1).$$

This would identify the first-return and return probabilities up to a constant along admissible times. The paper gives examples where the conjecture holds and proves weaker lower bounds for graphs with a closed, transitive, nonunimodular subgroup of automorphisms; the general assertion remains open.