Moduli-space conjecture for Kähler manifolds with complexified polarisation

Let $X$ be a Kähler manifold and let $[\omega^{\mathbb{C}}]$ be a complexified polarisation. For a coupling constant $\gamma$, consider pairs $(X,[\omega^{\mathbb{C}}])$ for which the cscK equation with B-field is solvable in the class $[\omega^{\mathbb{C}}]$.

Moduli-space conjecture. There exists a Hausdorff complex space $\mathfrak{M}_{\gamma}(X,[\omega^{\mathbb{C}}])$ serving as a moduli space of Kähler manifolds with a complexified polarisation, such that the cscK equation with B-field is solvable in the class $[\omega^{\mathbb{C}}]$. Moreover, $\mathfrak{M}_{\gamma}(X,[\omega^{\mathbb{C}}])$ admits a natural Weil–Petersson-type Kähler metric $\eta$.

This is proposed by analogy with classical moduli constructions for Kähler manifolds and with more recent moduli results. The existence of the asserted Hausdorff complex space and its natural Weil–Petersson-type metric remains open in the source.