Fano-type conjecture for covariant Kähler structures on quantum flag manifolds

Let ${\mathbb C}_q[G_0/L_0]$ be an irreducible quantum flag manifold. Let $(\Omega^{(\bullet,\bullet)},\kappa)$ consist of the Heckenberger–Kolb complex structure and a left-coinvariant form in $\Omega^{(1,1)}$, unique up to scalar multiple. Fano-type conjecture. For every irreducible quantum flag manifold ${\mathbb C}_q[G_0/L_0]$, the pair $(\Omega^{(\bullet,\bullet)},\kappa)$ is a covariant Kähler structure for the Heckenberger–Kolb calculus, and the associated complex structure is of Fano type. This is presented as a conjectural extension of the known quantum Grassmannian case to all irreducible quantum flag manifolds; the source gives no resolution status.