The Geometry of the affine closure of $T^*(\mathrm{SL}_n/U)$

We show that the affine closure $\overline{T^*(\mathrm{SL}_n/U)}$ has symplectic singularities, in the sense of Beauville. In the special case $n=3$, we show that the affine closure $\overline{T^*(\mathrm{SL}_3/U)}$ is isomorphic to the closure $\overline{\mathcal{O}}_\textrm{min}$ of the minimal nilpotent orbit $\mathcal{O}_{\textrm{min}}$ in $\mathfrak{so}_8$. Moreover, the quasi-classical Gelfand-Graev action of the Weyl group $W$ on $\overline{T^*(\mathrm{SL}_3/U)}$ can be identified with the restriction to $\overline{\mathcal{O}}_\textrm{min}$ of the triality $S_3$-action on $\mathfrak{so}_8$.