Asymptotic purity characterization for smooth projective threefolds

Let $X$ be a smooth, projective $3$-fold. A smooth projective variety is asymptotically pure (AP) if, for every divisor class, at most one asymptotic cohomological function is nonzero. The big cone is the cone of numerical divisor classes whose associated divisors are big, and the ample cone is the cone of numerical classes of ample divisors.

Asymptotic purity conjecture. $X$ is AP if and only if the big and ample cones are equal.

The conjecture seeks necessary and sufficient conditions for asymptotic purity. Equality of the big and ample cones is necessary for AP, but the converse for smooth projective threefolds is presented as the conjectural part.