Positivity conjecture for the Hull–Strominger moduli-space metric

Let $(X,P)$ be a Calabi–Yau threefold with bundle $P$ admitting a solution of the Hull–Strominger system. Let $\dot{\mathfrak{a}}\in H^{1,1}_A(X)$ and $\dot{\mathfrak{b}}\in H_{BC}^{2,2}(X)$ denote the complexified variations of the Aeppli and balanced classes of nearby solutions, and let $\mathfrak{b}$ be the balanced class of the solution. Positivity conjecture. The fibrewise moduli-space metric is positive definite. In particular, the variations satisfy

$$\operatorname{Re}\dot{\mathfrak{a}}\cdot\operatorname{Re}\dot{\mathfrak{b}}<\frac{1}{2\int_X\|\Omega\|_\omega\frac{\omega^3}{6}}\left(\operatorname{Re}\dot{\mathfrak{a}}\cdot\mathfrak{b}\right)^2.$$

This is a physical prediction for the moduli space of Hull–Strominger solutions, asserting positivity of the metric along the fibres and constraining variations of the Aeppli and balanced classes. The source presents it as a prediction rather than a proved result.