Hacon–Kovács–Luo–Zhang conjecture on zeros of holomorphic one-forms

Let $X$ be a smooth complex projective variety of general type. For a global holomorphic one-form $\omega\in H^0(X,\Omega_X^1)$, define its zero locus by

$$Z(\omega)=\{x\in X\mid \omega(T_xX)=0\}.$$

Hacon–Kovács–Luo–Zhang conjecture. The zero locus $Z(\omega)$ is nonempty for every global holomorphic one-form $\omega\in H^0(X,\Omega_X^1)$.

The conjecture concerns the existence of zeros of holomorphic one-forms on varieties of general type. It is tautological for curves, and was known for surfaces, threefolds, varieties with ample canonical bundle, and more generally minimal varieties; the supplied text does not establish that the conjecture is solved in full generality.