Collins–Jacob–Yau hypercritical-phase conjecture

Let $(X^n,\omega)$ be a connected, compact Kähler manifold, let $\Omega\in \mathrm{H}^{1,1}(X,\mathbb{R})$, and let $Z_V(\Omega)$ be the central charge associated with an irreducible subvariety $V\subset X$. Let $\mathcal{H}_{\Omega}$ denote the space of admissible representatives of $\Omega$.

Collins–Jacob–Yau hypercritical-phase conjecture. The set $\mathcal{H}_{\Omega}$ is non-empty and $\Omega$ has hypercritical phase if and only if, for every irreducible subvariety $V$ of $X$,

$$\operatorname{Im}\bigl(Z_V(\Omega)\bigr)>0.$$

This conjecture characterizes hypercritical classes through positivity of their central charges on all irreducible subvarieties. The supplied text presents it as a further conjecture of Collins–Jacob–Yau and cites later discussion, but gives no resolution status.