Lagrangian subvarieties of hyperspherical varieties related to $G_2$

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.