Boucksom's Fujiki class conjecture for positive currents

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

$$\int_X T^n_{ac}>0,$$

then $X$ belongs to the Fujiki class, meaning that $X$ is bimeromorphic to a compact Kähler manifold. This conjecture asks whether positivity of the volume associated with a closed positive current forces a compact complex manifold to have the birational Kähler property; the source gives no resolution.