Smoothness criterion for morphisms to simple abelian varieties

Let $f\colon X\to A$ be a morphism from a smooth projective variety $X$ to a simple abelian variety $A$. A holomorphic $1$-form on $A$ is an element of $H^0(A,\Omega_A^1)$.

Smoothness criterion. The morphism $f$ is smooth if and only if there is a holomorphic $1$-form $\omega\in H^0(A,\Omega_A^1)$ such that $f^*\omega$ has no zero.

This conjecture concerns the relationship between smoothness of a morphism to a simple abelian variety and the nonvanishing of a pulled-back holomorphic $1$-form. It is mentioned implicitly in the cited work of Dutta, Hacon and Liu and supported by results of Dutta, Hacon and Liu and of Schnell and Yang; its general status remains open.