The Hermitian structure conjecture for irreducible quantum flag manifolds

Let $\mathbb{C}_q[G_0/L_0]$ be an irreducible quantum flag manifold, with its Heckenberger--Kolb differential calculus $\Omega^{(\bullet,\bullet)}$ and the left-coinvariant form $\kappa$ in $\Omega^{(1,1)}$. Hermitian structure conjecture. For every irreducible quantum flag manifold $\mathbb{C}_q[G_0/L_0]$, the pair $(\Omega^{(\bullet,\bullet)},\kappa)$ is a covariant Hermitian structure for the Heckenberger--Kolb calculus. This conjecture proposes a uniform covariant Hermitian structure for the Heckenberger--Kolb calculi on irreducible quantum flag manifolds; the supplied text does not state whether it has been resolved.