The numerical inequality for irregular compact Kähler manifolds without irregular fibrations

Let $X$ be an irregular compact Kähler manifold with no irregular fibrations. Write $q(X)$ for its irregularity, $\dim X$ for its dimension, and $\chi(\omega_X)$ for the Euler characteristic of its canonical bundle.

Numerical inequality conjecture. If

$$\chi(\omega_X)\geq 2,$$

then

$$\chi(\omega_X)>q(X)-\dim X$$

when $q(X)$ is very large compared to $\chi(\omega_X)$.

The surrounding discussion presents this as an expected rarity phenomenon for irregular compact Kähler manifolds without irregular fibrations. The precise meaning of “very large compared to” is not specified in the source, and no resolution is given here.