Collins-J-Yau conjecture for the deformed Hermitian-Yang-Mills equation

Let $(X,\omega)$ be a compact Kähler manifold and let $[\alpha]\in H^{1,1}(X,\mathbb R)$. For an analytic subvariety $V\subseteq X$, define

$$Z_{[\alpha][\omega]}(V)=-\int_V e^{-i\omega+\alpha},$$

where only the term of order $\dim(V)$ in the expansion is integrated. The number $Z_{[\alpha][\omega]}(V)$ depends only on the cohomology classes $[\alpha]$ and $[\omega]$. The phase is supercritical when the constant $\hat\theta$ can be lifted to $\mathbb R$ and lies in $(((n-2)\frac{\pi}{2},n\frac{\pi}{2}))$.

Collins-J-Yau conjecture. The class $[\alpha]$ admits a solution to the deformed Hermitian-Yang-Mills equation with supercritical phase if and only if $Z(X)\neq 0$ and, for every analytic subvariety $V\subset X$,

$$\operatorname{Im}\left(\frac{Z_{[\alpha][\omega]}(V)}{Z_{[\alpha][\omega]}(X)}\right)>0.$$

This conjecture proposes that the necessary subvariety inequalities obtained by integrating the positivity condition are also sufficient for solvability. The source presents it as an open conjecture under the supercritical-phase assumption.