Atypicality and rank support varieties for stable polar Lie superalgebras

Let ${\mathfrak{g}}$ be a simple basic classical Lie superalgebra, and let $L(\lambda)$ be a finite-dimensional simple ${\mathfrak{g}}$-supermodule. Write $\operatorname{atyp}(L(\lambda))$ for its atypicality, and let ${\mathcal V}^{\operatorname{rank}}_{{\mathfrak e}}(L(\lambda))$ denote its rank support variety. Atypicality–support conjecture. If ${\mathfrak{g}}$ is stable and polar, then

$$\operatorname{atyp}(L(\lambda))=\dim {\mathcal V}^{\operatorname{rank}}_{{\mathfrak e}}(L(\lambda)).$$

The conjecture relates the highest-weight-theoretic atypicality of a simple module to the geometric dimension of its rank support variety. It is verified for Lie superalgebras of defect one, for modules of atypicality zero, and in one direction for ${\mathfrak{gl}}(m|n)$; the general stable polar case remains open.