K-stability criterion for solutions of the complexified equation

Let $(X,L)$ be a polarized complex variety, and let $\eta\in H^1(X,\mathcal{E}_L)$ be a deformation class of $(X,L)$. A test configuration is compatible with $\eta$ if its first-order deformation of the fibration is compatible with the relative polarization and restricts to $\eta$ on the general fibre. The K-stability conjecture. The equation in the paper admits a solution $\omega\in\mathrm{c}_1(L)$ with $\alpha(\omega)\in\eta$ if and only if $(X,L)$ is K-stable with respect to test configurations compatible with $\eta$. This is proposed as a stability characterization for the existence of solutions, extending the analogy with Higgs bundles; the general validity of the equivalence is left open.