  The generalized affine Grassmannian slices
$\overline{\mathcal{W}}_\mu^\lambda$ are algebraic varieties introduced by
Braverman, Finkelberg, and Nakajima in their study of Coulomb branches of $3d$
$\mathcal{N}=4$ quiver gauge theories. We prove a conjecture of theirs by
showing that the dense open subset $\mathcal{W}_\mu^\lambda \subseteq
\overline{\mathcal{W}}_\mu^\lambda$ is smooth. An explicit decomposition of
$\overline{\mathcal{W}}_\mu^\lambda$ into symplectic leaves follows as a
corollary. Our argument works over an arbitrary ring and in particular implies
that the complex points $\mathcal{W}_\mu^\lambda(\mathbb{C})$ are a smooth
holomorphic symplectic manifold.
