Esnault's finite-fundamental-group conjecture for Kähler manifolds

Let $X$ be a compact Kähler manifold. Let $\Omega_X^1$ denote its cotangent bundle, and let $H^0(X,\operatorname{Sym}^m\Omega_X^1)$ be the space of holomorphic symmetric $m$-differentials.

Esnault's conjecture. If

$$H^0(X,\operatorname{Sym}^m\Omega_X^1)=0$$

for all $m>0$, then the fundamental group of $X$ is finite.

The parser marks this conjecture as resolved. The source notes that a slightly weaker statement was known: under the same vanishing condition, the fundamental group has no linear representation with infinite image.