Luo–Zhang conjecture on zero-free holomorphic one-forms

Let $X$ be a smooth complex projective variety, let $\kappa(X)$ denote its Kodaira dimension, and let $W\subseteq H^0(X,\Omega_X^1)$ be a linear subspace such that the zero locus $Z(\omega)$ is empty for every nonzero one-form $\omega\in W$.

Luo–Zhang conjecture. The dimension of $W$ can be at most

$$\dim X-\kappa(X).$$

This is a generalization of the statement for varieties of general type, expressed in terms of Kodaira dimension. The source states that Luo and Zhang proved it for threefolds; its status in general is not resolved by the supplied text.