Boucksom-Demailly-P5un-Peternell duality conjecture for compact K4hler manifolds

Let $X$ be a compact K4hler manifold of complex dimension $n$. The cone $\mathcal{E}$ of pseudoeffective classes consists of classes in $H^{1,1}_{\mathbb{R}}(X)$ containing a closed positive current, and the movable cone $\mathcal{M}$ is the closed convex cone in $H^{n-1,n-1}_{\mathbb{R}}(X)$ generated by classes

$$\mu_*(\tilde{\beta}_1\wedge\cdots\wedge\tilde{\beta}_{n-1}),$$

where $\mu:\tilde{X}\to X$ is a smooth modification and the $\tilde{\beta}_i$ are K4hler classes on $\tilde{X}$. The Poincar9 pairing is $(\alpha\cdot\eta)=\int_X\alpha\wedge\eta$. Boucksom-Demailly-P5un-Peternell conjecture. On any compact K4hler manifold $X$, the cones $\mathcal{E}$ and $\mathcal{M}$ are dual via the Poincar9 pairing of $H^{1,1}_{\mathbb{R}}(X)$ with $H^{n-1,n-1}_{\mathbb{R}}(X)$. Equivalently, a class $\alpha\in H^{1,1}_{\mathbb{R}}(X)$ contains a closed positive current exactly when it has nonnegative pairing with every class in $\mathcal{M}$. This extends the projective-manifold duality theorem of Boucksom, Demailly, P5un and Peternell to the compact K4hler setting; the conjecture remains open in general.