  We consider two $S$-dual hyperspherical varieties of the group $G_2 \times
\text{SL}(2)$: an equivariant slice for $G_2$, and the symplectic
representation of $G_2 \times \text{SL}_2$ in the odd part of the basic
classical Lie superalgebra $\mathfrak{g}(3)$. For these varieties we check the
equality of numbers of irreducible components of their Lagrangian subvarieties
(zero levels of the moment maps of Borel subgroups' actions) conjectured in
arXiv:2310.19770.
