Projectivity conjecture for compact Kähler manifolds with a rational pseudoeffective dual class

Let $X$ be a compact Kähler manifold of dimension $n$. Let $\mathrm{Psef}(X)$ be the pseudoeffective cone in $H^{1,1}(X,\mathbf{R})$, and let $\mathrm{Int}(\mathrm{Psef}(X)^\vee)$ denote the interior of its Poincaré dual cone in $H^{n-1,n-1}(X,\mathbf{R})$. Projectivity conjecture. If

$$\mathrm{Int}(\mathrm{Psef}(X)^\vee) \cap H^{2n-2}(X,\mathbf{Q}) \ne \varnothing,$$

then $X$ is projective. This would give a positive answer to the stated dual problem to the Kodaira embedding theorem; the paper notes supporting evidence from threefolds and Ricci-flat manifolds, while the conjecture remains open in general.