Boucksom–Demailly–Păun–Peternell transcendental holomorphic Morse inequality conjecture

Let $X$ be a compact complex manifold of dimension $n$, and let $H^{1,1}_{BC}(X,\mathbb C)$ denote its complex Bott–Chern $(1,1)$-cohomology. A class is nef if it lies in the nef cone, and a Kähler current is a positive closed $(1,1)$-current that dominates a Hermitian form. The volume of a class is denoted by $\operatorname{Vol}$. Boucksom–Demailly–Păun–Peternell's conjecture. If $\alpha,\beta\in H^{1,1}_{BC}(X,\mathbb C)$ are nef classes satisfying

$$\alpha^n>n\alpha^{n-1}\cdot\beta,$$

then $\alpha-\beta$ contains a Kähler current and

$$\operatorname{Vol}(\alpha-\beta)\geq\alpha^n-n\alpha^{n-1}\cdot\beta.$$

This is proposed as a transcendental counterpart of the holomorphic Morse inequalities for integral classes. The supplied text gives no resolution of this conjecture, so its status remains open.