Mobility log-concavity conjecture

Let $X$ be a projective variety of dimension $n$. For a class in $N_k(X)$, let $\operatorname{mob}$ denote its mobility, and let $\operatorname{\overline{Eff}}_k(X)$ be the pseudo-effective cone of $k$-cycle classes. Mobility log-concavity conjecture. The function $\operatorname{mob}$ is log-concave on $\operatorname{\overline{Eff}}_k(X)$: for any classes $\alpha,\beta\in\operatorname{\overline{Eff}}_k(X)$,

$$\operatorname{mob}(\alpha+\beta)^{\frac{n-k}{n}}\geq \operatorname{mob}(\alpha)^{\frac{n-k}{n}}+\operatorname{mob}(\beta)^{\frac{n-k}{n}}.$$

This would generalize the log-concavity of the volume of divisors to arbitrary cycle classes. Its status is not resolved in the supplied text.