Lagrangian subvarieties of hyperspherical varieties related to G₂

We consider two S-dual hyperspherical varieties of the group G₂ × SL(2): an equivariant slice for G₂, and the symplectic representation of G₂ × SL₂ in the odd part of the basic classical Lie superalgebra 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.