The Kähler structure conjecture for irreducible quantum flag manifolds

Let $\mathcal{O}_q(G/L_S)$ be an irreducible quantum flag manifold, with its Heckenberger--Kolb calculus $\Omega^{(bullet,bullet)}$. A covariant Kähler structure consists of the pair $(\Omega^{(bullet,bullet)},\kappa)$. Kähler structure conjecture. For every irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$, a covariant Kähler structure for its Heckenberger--Kolb calculus is given by $(\Omega^{(bullet,bullet)},\kappa)$. The conjecture was originally proposed in the cited earlier work and would extend the Kähler geometry known for quantum projective space to all irreducible quantum flag manifolds. The source does not report a general proof.