Boucksom–Demailly–Păun–Peternell holomorphic Morse inequality conjecture

Let $X$ be a compact complex manifold of complex dimension $n$. Let $\alpha$ be a closed real $(1,1)$-form, and let $X(\alpha,\leq 1)$ be the set where $\alpha$ is nondegenerate and has at most one negative eigenvalue. Boucksom–Demailly–Păun–Peternell's conjecture. If

$$\int_{X(\alpha,\leq 1)}\alpha^n>0,$$

then the class $\{\alpha\}\in H^{1,1}_{\partial\overline{\partial}}(X,\mathbb{R})$ is big and

$$\operatorname{vol}(\{\alpha\}):=\sup_{0<T\in\{\alpha\}}\int_{X\setminus \operatorname{Sing}(T)}T^n\geq\int_{X(\alpha,\leq 1)}\alpha^n,$$

where $T$ ranges over all Kähler currents in $\{\alpha\}$ with analytic singularities. The conjecture proposes a transcendental holomorphic Morse inequality controlling the volume of a big class by the integral over the locus with at most one negative eigenvalue; the supplied source does not provide a resolution.