The Demailly–Păun conjecture on nef classes with positive volume

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. Let $\{\beta\}\in H^{1,1}(X,\mathbb{R})$ be a real $(1,1)$-class with smooth representative $\beta$. The Demailly–Păun conjecture. If $\{\beta\}$ is nef and

$$\int_X\beta^n>0,$$

then $\{\beta\}$ is big, meaning that it contains a Kähler current. This is the Hermitian analogue of the nef-and-big characterization familiar in the Kähler setting. The supplied material gives no resolution status for this formulation, so it is recorded as open.