The nilpotent quotient conjecture for nef Kähler manifolds

Let $M$ be a nef Kähler manifold of dimension $n$. Suppose there is an epimorphism

$$\varphi:\pi_1(M)\to\Gamma,$$

where $\Gamma$ is a torsion-free nilpotent group of rank at least $2n$. Nilpotent quotient conjecture. Then

$$\Gamma\cong\mathbb Z^{2n}$$

and $M$ is biholomorphic to the complex torus $T^n_{\mathbb C}$. This is proposed as a consequence of the conjectures concerning the fundamental groups of nef Kähler manifolds; its status is open in the source.