Monotonicity conjecture for simple random walk on growing graphs

Let $\bmathbb{G}_t$ and $\bmathbb{G}'_t$ be graphs non-decreasing in $t$, of uniformly bounded degrees, with $\bmathbb{G}_t\subseteq\bmathbb{G}'_t$ for every $t$. Let $\bmathbb{Y}_t$ and $\bmathbb{Y}'_t$ be simple random walks on these graph evolutions, both starting from $y_0$. Monotonicity conjecture. If $\bmathbb{Y}_t$ is transient, meaning that its sample path almost surely returns to $Y_0=y_0$ only finitely often, then $\bmathbb{Y}'_t$ is also transient. The conjecture is a time-dependent analogue of Rayleigh monotonicity, which gives the claim when the graphs are fixed. Its validity for general growing graphs remains open; the paper notes that bounded degrees are essential to the formulation.