The generalized LeBrun equality for compact complex manifolds

Let $M$ be a compact complex $n$-manifold, and let $K_M$ denote its canonical bundle. Define the canonical volume by $\mathrm{CanVol}(M)$ and the complex minimal volume and associated invariants by $\mathrm{MinVol}_C(M)$, $\mathcal{I}_C(M)$, and $\mathcal{I}_C^-(M)$. The equality established in the Kähler case with nef canonical bundle is

$$\mathrm{MinVol}_C(M)=\mathcal{I}_C(M)=\mathcal{I}_C^-(M)=\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M).$$

Generalized LeBrun equality. The equality above should hold for any compact complex $n$-manifold, at least when $K_M$ is big.

The claim extends the stated theorem for compact Kähler manifolds whose canonical bundle is nef; its validity beyond that setting, including the case of compact complex manifolds with big canonical bundle, remains open.