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We show that for any singular dominant integral weight $\lambda$ of a complexsemisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of anyprojective-injective module of the parabolic BGG category$\mathcal{O}_\lambda^{\mathfrak{p}}$ is a symmetric algebra (as conjectured byKhovanov) extending the results of Mazorchuk and Stroppel for the regulardominant integral weight. Moreover, the endomorphism algebra $B$ is equippedwith a homogeneous (non-degenerate) symmetrizing form. In the appendix, thereis a short proof due to K. Coulembier and V. Mazorchuk showing that theendomorphism algebra $B_\lambda^{\mathfrak{p}}$ of the basicprojective-injective module of $\mathcal{O}_\lambda^{\mathfrak{p}}$ is asymmetric algebra.

Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category