Kotschick's conjecture on nonvanishing one-forms on compact Kähler manifolds

Let $X$ be a compact Kähler manifold. A holomorphic one-form without zeros is a holomorphic $1$-form on $X$ that vanishes nowhere, and a real closed $1$-form without zeros is a real closed $1$-form on $X$ that vanishes nowhere.

Kotschick's conjecture. The following conditions are equivalent:

$$\text{(A) } X \text{ admits a holomorphic one-form without zeros};$$ $$\text{(B) } X \text{ admits a real closed }1\text{-form without zeros}.$$

The paper’s abstract states that the authors prove Kotschick’s conjecture in dimension $3$; the conjecture in arbitrary dimension is therefore not established here.