FMP stability conjecture for big movable classes

Let $X$ be a compact Kähler manifold of dimension $n$, and let $\alpha,\beta\in\operatorname{Mov}^1(X)$ be big movable classes. Let $F(\alpha,\beta)$ denote their relative asymmetry index, and let $\sigma(\alpha,\beta)$ and $B(\alpha,\beta)$ denote the relative size index and the Brunn–Minkowski deficit. FMP stability conjecture. There exists a constant $c(n)$ depending only on $n$ such that

$$F(\alpha,\beta)\leq c(n)\bigl(\sigma(\alpha,\beta)B(\alpha,\beta)\bigr)^{\frac12}.$$

This is the proposed Kähler-geometric analogue of the Fusco–Maggi–Pratelli and Figalli–Maggi–Pratelli stability estimate for convex bodies. Its validity for arbitrary big movable classes is left open.