The basic $\ell^2$-distance conjecture for Ramanujan graphs

Let $X$ be a sequence of Ramanujan graphs, let $n$ denote the number of vertices, let $d$ be their common degree, and set $p=d-1$. Let $d_2(t)$ be the average squared $\ell^2$ distance from stationarity at time $t$, and let $N(t)$ denote the quantity defined in the paper. The basic $\ell^2$-distance conjecture. If $t<2 ot\log_p n$, then

$$d_2(t)\sim\frac{1}{N(t)}$$

as $n\to\infty$. This conjecture predicts the asymptotic behavior of the distance before the cutoff window for non-backtracking random walks on Ramanujan graphs; the supplied text does not state whether it has been proved or disproved.