Probabilistic formula for the energy-kernel representative on infinite resistance networks

Let $G$ be an infinite resistance network with origin $o$, and let $v_x$ be the representative of an element of the energy kernel normalized by $v_x(o)=0$. Write $R^F(o,x)$ for the free effective resistance between $o$ and $x$, let $\tau_z$ denote the hitting time of $z$, and let $|\mathcal{C}|<\infty$ denote the event that the random-walk trajectory is bounded, meaning that it is contained in a finite subnetwork of $G$. Probabilistic formula for the energy-kernel representative. For $x\ne o$, the function $v_x$ is given by

$$v_x(y)=R^F(o,x)\,\mathbb{P}_y[\tau_x<\tau_o\mid |\mathcal{C}|<\infty].$$

Thus the random walk is conditioned to lie entirely in some finite subnetwork. The conjecture proposes an extension of the corresponding probabilistic formula from finite resistance networks to infinite ones; the source explicitly presents the accompanying argument as an erroneous proof, so the formula remains unverified here.