Cone duality conjecture for compact balanced manifolds

Let $X$ be a compact balanced manifold. Let $\mathcal E$ denote the relevant cone of classes on $X$, let $\mathcal B$ denote the balanced cone, and let $\mathcal E^\vee$ be the dual cone. Write $\overline{\mathcal B}$ for the closure of the balanced cone.

Cone duality conjecture. The dual cone of $\mathcal E$ is the closure of the balanced cone:

$$\mathcal E^\vee=\overline{\mathcal B}.$$

This proposes the cone duality known in the cited special cases for every compact balanced manifold, extending the relationship between movable and balanced classes discussed immediately before the statement. Its general validity is left as a problem.