Concavity conjecture for the variation of cycle classes

Let $X$ be an integral projective variety, and let $N_k(X)_{\mathbb{Z}}$ denote the group of numerical classes of $k$-cycles. A class is pseudo-effective if it lies in the pseudo-effective cone, and a class is big if it lies in its interior. For a class $\alpha$, write $\operatorname{chdim}(\alpha)$ for its Chow dimension. Variation concavity conjecture. If $\alpha,\beta,\gamma\in N_k(X)_{\mathbb{Z}}$ with $\alpha$ pseudo-effective and $\beta$ and $\gamma$ big, then

$$\operatorname{chdim}(\alpha + \beta) - \operatorname{chdim}(\alpha) \leq \operatorname{chdim}(\alpha + \beta + \gamma) - \operatorname{chdim}(\alpha + \gamma).$$

This is a weak concavity statement for the variation of cycle classes, analogous to concavity properties of the volume. The source notes that it would imply upper semicontinuity of the variation on the pseudo-effective cone; its resolution is not specified here.