Boucksom's transcendental bigness conjecture

Let $X$ be a compact complex manifold of complex dimension $n$, and let $T$ be a closed positive $(1,1)$-current on $X$. Write $T_{ac}$ for its absolutely continuous part and $\{T\}$ for its cohomology class. Boucksom's conjecture. If

$$\int_X T_{ac}^n>0,$$

then the class $\{T\}$ is big, and consequently $X$ belongs to the Fujiki class $\mathcal{C}$. The conjecture extends Grauert–Riemenschneider-type criteria from integral classes to transcendental classes; the supplied source says it is proved in the integral case and attributes the general conjecture's proof to Boucksom.