Characterization of torus quotients

Let $X$ be a compact complex space of dimension $n\geq 2$. Let $\widetilde{c}_2(X)$ denote the second orbifold Chern class of $X$, and let a finite group action be free in codimension one when its non-free locus has codimension at least two. Characterization of torus quotients. The following are equivalent:

1. $X$ has klt singularities, $c_1(X)=0\in H^2(X,\mathbb{R})$, and there exists a Kähler class $\omega\in H^2(X,\mathbb{R})$ such that

$$\widetilde{c}_2(X)\cdot\omega^{n-2}=0.$$

2. There exists a complex torus $T$ and a holomorphic action of a finite group $G$ on $T$, free in codimension one, such that

$$X\cong T/G.$$

In dimension $2$, this characterization is well known. In dimension $3$, it would follow from the stated three-dimensional result together with the relevant special case of the Abundance Conjecture; the general statement in dimensions $n\geq 2$ remains open.