Campana's special-manifold conjecture S

Let $X$ be a special compact Kähler manifold. Its $\gamma$-dimension $\gamma d(X)$ is the dimension of the base of the canonical almost holomorphic map associated with the maximal subspaces whose fundamental groups have finite image in $\pi_1(X)$. Assume

$$\gamma d(X)=\dim(X).$$

Conjecture S. Some finite étale cover of $X$ is bimeromorphic to a complex torus.

This is an Iitaka-type conjecture for special manifolds of $\pi_1$-general type. The source presents it in the context of the relationship between abelianity, Iitaka's conjecture, and Campana's special manifolds; no general resolution is given.