Almost-sure convergence of delayed random walk on ergodic unimodular random graphs

Let $(G,o)$ be an ergodic unimodular random graph of bounded degrees, and let $(X_n)$ be delayed random walk started from $o$.

Delayed random-walk convergence conjecture. The distribution of $(G,X_t)$ converges to the initial unimodular measure almost surely.

This question arises from the proof of triviality of an invariant sigma-field for a labelled graph and is stated in the source as apparently open. The claim concerns the almost-sure convergence of the environment viewed from delayed random walk.