Collins–Yau's Nakai–Moishezon criterion for the dHYM equation

Let $(X,c9)$ be a compact Kähler manifold, let $[\alpha]\in H^{1,1}(X,\mathbb{R})$, let $\mathcal{H}_{\omega}$ denote the space of solutions to the deformed Hermitian–Yang–Mills equation in the class $[\alpha]$, and let $Z_{V,[\alpha]}$ be the central charge associated with an irreducible subvariety $V\subset X$. The class $[\alpha]$ has hypercritical phase when its dHYM phase lies in the hypercritical range.

Collins–Yau's conjecture. The following are equivalent:

1. $\mathcal{H}_{\omega}$ is non-empty and $[\alpha]$ has hypercritical phase.
2. For every irreducible subvariety $V\subset X$,

$$\operatorname{Im}(Z_{V,[\alpha]})>0.$$

The conjecture is a Nakai–Moishezon type criterion for solvability of the dHYM equation. The supplied text reports a counterexample on $X=\operatorname{Bl}_{p}(\mathbb{CP}^{2})$, so the equivalence is disproved.