Khovanov's symmetry conjecture for endomorphism algebras in parabolic category O

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with fixed Borel subalgebra $\mathfrak{b}$, Cartan subalgebra $\mathfrak{h}\subset\mathfrak{b}$, and parabolic subalgebra $\mathfrak{p}\supset\mathfrak{b}$. For a dominant integral weight $\lambda$, let $\mathcal{O}^\mathfrak{p}_\lambda$ be the corresponding block of parabolic category $\mathcal{O}$. A projective-injective module is a module that is both projective and injective in this category. Khovanov's conjecture. For every projective-injective module $P$ in $\mathcal{O}^\mathfrak{p}_\lambda$, the algebra

$$\operatorname{End}_{\mathcal{O}^\mathfrak{p}_\lambda}(P)$$

is symmetric. The claim concerns the endomorphism algebras of self-dual projective modules in parabolic category $\mathcal{O}$; the supplied context records it as a speculation attributed to Khovanov, but gives no resolution status.