The spectral triple conjecture for irreducible quantum flag manifolds

Let $\mathbb{C}_q[G/L_S]$ be an irreducible quantum flag manifold, and let $\Omega^{(0,\bullet)}$ be the subcomplex of its Heckenberger--Kolb calculus. Denote by $L^2(\Omega^{(0,\bullet)})$ the completion with respect to the inner product associated to the Hermitian form $\kappa$, and let $\overline{\partial}^{\dagger}$ be the adjoint of $\overline{\partial}$. Spectral triple conjecture. A spectral triple is given by

$$\left(\mathbb{C}_q[G/L_S], L^2(\Omega^{(0,\bullet)}), \overline{\partial}+\overline{\partial}^{\dagger}\right).$$

This conjecture seeks to complete the conjectured Kähler structures on irreducible quantum flag manifolds to spectral triples. The case of quantum projective space $\mathbb{C}_q[\mathbb{C}P^n]$ is treated in the cited literature, while the general assertion is not resolved in the supplied text.