Singular J-equation conjecture for normal Kähler varieties

Let $X$ be an $n$-dimensional normal Kähler variety with Kähler classes $\alpha$ and $\beta$. The pair $(\alpha,\beta)$ is $J$-positive when, for every $m$-dimensional analytic subvariety $Z\subset X$,

$$m\left.\frac{\alpha^{m-1}\cdot\beta}{\alpha^m}\right|_Z<n\frac{\alpha^{n-1}\cdot\beta}{\alpha^n}.$$

For a Kähler form $\chi\in\beta$, consider the $J$-equation

$$\omega^{n-1}\wedge\chi=\left(\frac{\alpha^{n-1}\cdot\beta}{\alpha^n}\right)\omega^n.$$

Singular J-equation conjecture. If $(\alpha,\beta)$ is $J$-positive, then this equation admits a unique solution $\omega$ as a Kähler current with bounded local potentials.

This extends the smooth solvability criterion to normal Kähler varieties. The source proposes it as an extension of the Lejmi–Szekelyhidi conjecture; its resolution is not established in the supplied text.