  We show that if $X$ is a normal complex quasi-projective variety, the
quasi-Albanese map of which is proper, then the torsionfree nilpotent quotients
of $\pi_1(X)$ are, up to a controlled finite index, the same ones as those of
the normalisation of its quasi-Albanese image. When $X$ is quasi-K\"ahler
smooth, we get the same conclusion, but only for the smooth models of the
quasi-Albanese image. In this second case, the proof is elementary, as the one
given in [Cam95] for $X$ compact.
  In the normal quasi-projective case, the \'etale Galois cover of $X$
associated to the nilpotent completion of $\pi_1(X)$ is thus holomorphically
convex. This is proved in the smooth case by $3$ other methods in [GGK22],
which motivated the present text.
  When $X$ is `special' in the sense of [Cam11], we deduce that the torsion
free nilpotent quotients of $\pi_1(X)$ are abelian. Examples show that this
property fails (as first observed in [CDY22]) when the quasi-Albanese map is
not proper. This leads to replace our previous `Abelianity conjecture' in the
compact case by an `Nilpotency conjecture' in the non-compact quasi-K\"ahler
context.
