Kähler-potential conjecture for the Hull–Strominger moduli space

Let $(X,\Omega)$ be a fixed Calabi–Yau threefold and let $P$ be a fixed bundle. Consider the moduli space of solutions of the Hull–Strominger system, and define

$$K=-\log\int_X\|\Omega\|_\omega\frac{\omega^3}{6}.$$

Kähler-potential conjecture. The function $K$ defines the Kähler potential for a Kähler metric on this moduli space.

The conjecture is motivated by the physical prediction that the moduli-space metric is positive definite along the fibres. The source does not provide a proof or a resolution.