Schreieder–Kotschick equivalence conjecture for holomorphic 1-forms

Let $X$ be a smooth complex projective variety. A global holomorphic 1-form without zeros means an element of $H^0(X,\Omega_X^1)$ whose zero locus is empty. A smooth fibre bundle structure over the circle means a smooth locally trivial fibration from $X$ to the circle; equivalently, $X$ admits a smooth real closed 1-form without zeros. For a finite étale morphism $\tau\colon X'\to X$ and $\omega\in H^0(X,\Omega_X^1)$, consider the complex

$$(H^{\bullet}(X',\mathbb{C}),\wedge\tau^*\omega).$$

Schreieder–Kotschick conjecture. The following three statements are equivalent: (1) $X$ admits a global holomorphic 1-form without zeros; (2) $X$ admits a smooth real closed 1-form without zeros, or equivalently a smooth fibre bundle structure over the circle; (3) there exists $\omega\in H^0(X,\Omega_X^1)$ such that, for every finite étale morphism $\tau\colon X'\to X$, the complex $(H^{\bullet}(X',\mathbb{C}),\wedge\tau^*\omega)$ is exact. This conjecture connects the existence of nowhere-vanishing holomorphic forms with the topology of Albanese maps and generic vanishing phenomena. The paper studies the equivalence and establishes related results, but the supplied text does not state that the conjecture is resolved.