Hall–Rado criterion for positive products of movable classes

Let $X$ be a compact Kähler manifold of dimension $n$, and let $L_1,\ldots,L_k$ be movable classes, where $1\leq k\leq n$. For $I\subseteq [k]$, write $L_I=\sum_{i\in I}L_i$, and let $\operatorname{nd}$ denote numerical dimension. Hall–Rado conjecture for movable classes.

$$\left\langle L_1\cdot\ldots\cdot L_k\right\rangle\neq 0 \quad\Longleftrightarrow\quad \operatorname{nd}(L_I)\geq |I|\quad\text{for all }I\subseteq [k].$$

This conjecturally extends the Hall–Rado positivity criterion from collections with nefness assumptions to arbitrary movable collections, replacing the ordinary intersection product by the positive product.