Demailly's transcendental Morse inequalities conjecture

Let $X$ be an $n$-dimensional compact complex manifold, and let $\alpha$ be a real $d$-closed $C^\infty$ $(1,1)$-form on $X$. Define

$$X(\alpha,\leq 1)=\{x\in X:\alpha\text{ has at most one negative eigenvalue at }x\}.$$

The Bott\Chern cohomology class $\{\alpha\}_{BC}$ is the class of $\alpha$ in Bott\Chern cohomology. Demailly's transcendental Morse inequalities conjecture. If

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

then $\{\alpha\}_{BC}$ contains a Kähler current (a closed positive $(1,1)$-current dominating a positive multiple of a Hermitian form). In particular, $X$ is a class $\mathcal{C}$ manifold, meaning that it is bimeromorphic to a Kähler compact complex manifold. This conjecture extends holomorphic Morse inequalities from integral $(1,1)$-classes represented by line bundles to possibly transcendental cohomology classes; its resolution would give a broad criterion for a compact complex manifold to belong to Fujiki's class $\mathcal{C}$.