Lejmi–Szekelyhidi's numerical criterion for the J-equation

Let $X$ be an $n$-dimensional compact Kähler manifold with Kähler classes $\alpha$ and $\beta$ satisfying the normalization condition

$$\alpha^n=\alpha^{n-1}\cdot\beta.$$

For a Kähler form $\chi\in\beta$, the $J$-equation for a Kähler form $\omega\in\alpha$ is

$$\omega^n=\omega^{n-1}\wedge\chi.$$

Lejmi–Szekelyhidi's conjecture. Under this normalization, the $J$-equation admits a unique smooth solution $\omega$ if and only if, for every $m$-dimensional analytic subvariety $Z\subset X$ with $1\le m\le n-1$,

$$\left(n\alpha^m-m\alpha^{m-1}\cdot\beta\right)\cdot Z=\int_Z\left(n\alpha^m-m\alpha^{m-1}\wedge\beta\right)>0.$$

This is the proposed holomorphic/topological criterion replacing the pointwise subsolution condition for the $J$-equation. The paper states that the criterion is subsequently proved through the equivalent notion of $J$-positivity, so the conjecture is presented in the source as a resolved result.