FMP stability conjecture on a Lorentzian class

Let $X$ be a compact Kähler manifold, let $\Omega$ be a Lorentzian class of dimension $d$ on $X$, and let $\alpha,\beta$ be big nef classes on $\Omega$. Define $F_\Omega(\alpha,\beta)$ by

$$F_\Omega(\alpha,\beta)=1-\sup_{\gamma\leq r\alpha,\gamma\leq\beta}\frac{|\gamma|_\Omega}{|\beta|_\Omega},$$

where the supremum ranges over big nef classes $\gamma$ satisfying $\gamma\leq r\alpha$ and $\gamma\leq\beta$, and $A\leq B$ means

$$(B-A)\cdot B_1\cdot\ldots\cdot B_{d-1}\cdot\Omega\geq0$$

for every nef class $B_i$. Let $\sigma(\alpha,\beta)$ be the relative size on $\Omega$, and let $B(\alpha,\beta)$ be the Brunn–Minkowski deficit. Lorentzian FMP conjecture. There exists a constant $c(d)$ depending only on $d$ such that

$$F_\Omega(\alpha,\beta)\leq c(d)\sqrt{\sigma(\alpha,\beta)B(\alpha,\beta)}.$$

This is the proposed Lorentzian-class analogue of the FMP stability estimate. The source gives no resolution and leaves the conjecture open.