The random walk on upper triangular matrices over $\mathbb{Z}\mathrm{/}m\mathbb{Z}$

We study a natural random walk on the $n\times n$ upper triangular matrices, with entries in $\mathbb{Z}\mathrm{/}m\mathbb{Z}$, generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this random walk is $O(m^{2}n\log n+n^2{m}^{o(1)})$. This answers a question of Stong and of Arias-Castro, Diaconis, and Stanley.