Infinite fundamental group under weak semiampleness and intermediate Kodaira dimension

Let $X$ be a compact complex manifold, let $\dim X$ denote its complex dimension, and let $k(X)$ denote its Kodaira dimension. Let $\Omega_X^1$ be the cotangent bundle, assumed to be weakly semiample.

Infinite fundamental-group conjecture. If

$$\dim X<k(X)$$

then the fundamental group of $X$ is infinite.

The paper explains that the statement follows in the Kähler case from the Wu–Zheng conjecture below, while it is presented as open in the general compact complex setting.