Collins–Jacob–Yau's stability conjecture for the dHYM equation

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, let $\alpha^{\mathbb{C}}=\alpha+\mathrm{i}\beta$ be a complexified Kähler class, and let $\hat{\vartheta}$ be its phase. Assume that the $(1,1)$-class $\beta$ has supercritical phase. For a proper irreducible analytic subvariety $V\subset X$ of dimension $p$, write $\iota_V:V\hookrightarrow X$ for the inclusion. Collins–Jacob–Yau's stability conjecture. There is a solution $B\in\beta$ of the dHYM equation if and only if, for every proper irreducible analytic subvariety $V\subset X$ with $1\leq p<n$,

$$\operatorname{Im}\left(\mathrm{e}^{-\mathrm{i}\hat{\vartheta}}(\iota_V^*\alpha^{\mathbb{C}})^p\right)<0.$$

This conjecture proposes a numerical criterion for solvability of the deformed Hermitian Yang–Mills equation, analogous to stability criteria in the setting of the Calabi conjecture and related geometric PDEs. Its general status is not established in the supplied source.