Submodularity conjecture for numerical dimensions of nef and movable classes

Let $X$ be a compact Kähler manifold. For a nef class $L$, write $\operatorname{nd}(L)$ for its numerical dimension, and let $M$ be another nef class and $\alpha$ a movable class. Submodularity conjecture. One has

$$\operatorname{nd}(L+M+\alpha)-\operatorname{nd}(L+M)\leq \operatorname{nd}(L+\alpha)-\operatorname{nd}(L).$$

The conjecture is motivated by an annihilation theorem asserting that failure of a movable class to increase numerical dimension persists after adding another nef class. The source also indicates that a broader submodularity statement for two merely movable classes is conjectural.