Lagrangian subvarieties of hyperspherical varieties

Given a hyperspherical $G$-variety $\mathcal{X}$ we consider the zero moment level ${\mathrm{\Lambda }}_{\mathcal{X}}\subset \mathcal{X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that ${\mathrm{\Lambda }}_{\mathcal{X}}$ is Lagrangian. For the dual $G^{\vee }$-variety ${\mathcal{X}}^{\vee }$, we conjecture that that there is a bijection between the sets of irreducible components $\mathrm{I}\mathrm{r}\mathrm{r}{\mathrm{\Lambda }}_{\mathcal{X}}$ and $\mathrm{I}\mathrm{r}\mathrm{r}{\mathrm{\Lambda }}_{{\mathcal{X}}^{\vee }}$. We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.