Symmetry conjecture for projective-injective modules of parabolic category O

Let $\mathfrak{g}$ be an arbitrary quasi-reductive Lie superalgebra and let $\mathfrak p$ be a parabolic subalgebra of $\mathfrak{g}$. Write $\mathcal{PI}^{\mathfrak p}$ for the full subcategory of the parabolic category $\mathcal O^{\mathfrak p}$ consisting of projective-injective objects, and let $S^{\rm top}(\mathfrak{g}_{\bar 1})$ denote the top symmetric power of the odd part, regarded as a $\mathfrak{g}$-module. Symmetry conjecture for projective-injective modules. The category $\mathcal{PI}^{\mathfrak p}$ is symmetric if and only if $S^{\rm top}(\mathfrak{g}_{\bar 1})$ is isomorphic to the trivial $\mathfrak{g}$-module.

The conjecture extends results known for basic classical and Q-type Lie superalgebras under additional assumptions, and for general linear Lie superalgebras. It predicts that symmetry is characterized precisely by triviality of the top symmetric power of the odd part.