Aldous–Diaconis universality of cutoff conjecture for random Cayley graphs

Let $G$ be a group, let $k$ satisfy $k \gg \log |G|$ and $\log k \ll \log |G|$, and let $\mathcal{G}_k$ be the Cayley graph of $G$ generated by $k$ independently and uniformly chosen random elements of $G$. Aldous and Diaconis's universality of cutoff conjecture. The random walk on $\mathcal{G}_k$ exhibits cutoff with high probability. This conjecture concerns the universality of the cutoff phenomenon for random walks on random Cayley graphs. The supplied context does not state whether it has been resolved.