The Dirac–Dolbeault spectral triple conjecture for weak Gelfand-type quantum flag manifolds

Let $\mathcal{O}_q(G/L_S)$ be an irreducible quantum flag manifold of weak Gelfand type, and let $(\Omega^{(bullet,bullet)},\kappa)$ be its covariant Kähler structure, unique up to real scalar multiple. Dirac–Dolbeault spectral triple conjecture. There is a Dirac–Dolbeault pair of spectral triples

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

and the associated $K$-homology class of each spectral triple is non-trivial. The conjecture is motivated by preliminary investigations suggesting solidity, which is equivalent here to compact resolvent for the Dolbeault--Dirac operator. The source restricts the assertion to weak Gelfand-type spaces and does not report a resolution.