Atypicality conjecture for support variety dimensions

Let ${\mathfrak{g}}$ be a simple basic classical Lie superalgebra with a non-degenerate invariant supersymmetric even bilinear form, and let $L(\lambda)$ be a finite dimensional simple ${\mathfrak{g}}$-supermodule. The atypicality of $L(\lambda)$, denoted $\operatorname{atyp}(L(\lambda))$, is the maximal number of linearly independent, mutually orthogonal, positive isotropic roots $\alpha\in\Phi^+$ satisfying $(\lambda+\rho,\alpha)=0$, where

$$\rho=\frac{1}{2}\left(\sum_{\alpha\in\Phi^+_{\bar 0}}\alpha-\sum_{\alpha\in\Phi^+_{\bar 1}}\alpha\right).$$

Atypicality conjecture.

$$\operatorname{atyp}(L(\lambda))=\dim {\mathcal V}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar 0})}(L(\lambda)).$$

This modifies the earlier atypicality conjecture by expressing the expected equality using the support variety for the pair $({\mathfrak{g}},{\mathfrak{g}}_{\bar 0})$, rather than a detecting subalgebra. The corresponding statement was verified for ${\mathfrak g}=\mathfrak{gl}(m|n)$, while the general case for simple basic classical Lie superalgebras remains open.