  Braverman, Finkelberg and Nakajima introduced the generalized affine
Grassmannian slices $\overline{\mathcal W}^{\lambda}_{\mu}$ and showed that
they are Coulomb branches of $3d$ $\mathcal N=4$ gauge theories. We prove a
conjecture of theirs that a transversal slice of $\mathcal W^{\nu}_{\mu}$ in
$\overline{\mathcal W}^{\lambda}_{\mu}$ is isomorphic to $\overline{\mathcal
W}^{\lambda}_{\nu}$. In addition, they conjecture that Coulomb branches of $3d$
$\mathcal N=4$ gauge theories have symplectic singularities, and we confirm
this conjecture for generalized affine Grassmannian slices $\overline{\mathcal
W}^{\lambda}_{\mu}$. Along the way we give a new proof of the smoothness of
$\mathcal W^{\nu}_{\mu}$, which has been previously proven by Muthiah and
Weekes using different method.
