Pullback conjecture for basepoint-free classes and movable transforms

Let $X$ be a smooth variety and let $\alpha\in\operatorname{BPF}_k(X)$, where $\operatorname{BPF}_k(X)$ denotes the cone of basepoint-free $k$-cycle classes. Let $\pi:Y\to X$ be a birational morphism. Pullback conjecture for basepoint-free classes. The class $\pi^*\alpha$ is a movable transform for $\alpha$. This would identify pullbacks of basepoint-free classes as movable transforms under every birational morphism. Its status is not resolved in the supplied text.