Dolbeault–Dirac spectral triple conjecture for irreducible quantum flag manifolds

Let $\mathcal{O}_q(G/L_S)$ be an irreducible quantum flag manifold with $q\in\mathbb{R}$, equipped with its Heckenberger–Kolb calculus $\Omega^\bullet$, covariant complex structure $\Omega^{(\bullet,\bullet)}$, and Kähler form $\kappa$. Let $D_{\overline{\partial}}$ be the associated Dolbeault–Dirac operator, acting on $L^2\big(\Omega^{(0,\bullet)}\big)$, and let $\gamma$ denote the grading.

Dolbeault–Dirac spectral triple conjecture. The point spectrum of $D_{\overline{\partial}}$ has finite multiplicity and tends to infinity. Consequently,

$$\left(\mathcal{O}_q(G/L_S),L^2\big(\Omega^{(0,\bullet)}\big),D_{\overline{\partial}},\gamma\right)$$

is a Dolbeault–Dirac spectral triple with non-trivial associated $K$-homology class.

For quantum projective space, the point spectrum and the non-trivial $K$-homology class are known. Extending these properties from quantum projective space to all irreducible quantum flag manifolds is presented as a goal and remains open.