Numerical criterion for solvability of the deformed Hermitian–Yang–Mills equation

Let $(X,\omega)$ be a Kähler manifold, let $L$ be a line bundle or let $[\alpha]$ be a real $(1,1)$ class, and let $Z_V(L)$ denote the central charge associated to each analytic subvariety $V\subset X$. Let $\mathcal H$ denote the space of solutions to the deformed Hermitian–Yang–Mills equation. Numerical criterion conjecture. If $Z_X(L)\in\mathbb H$, then $\mathcal H\ne\emptyset$ if and only if $Z_V(L)\in\mathbb H$ for all $V\subset X$. This is proposed as a converse to the preceding numerical obstruction, analogous to the Nakai–Moishezon and Demailly–Păun criteria; the supplied status evidence does not establish a resolution.