  In this short note, we consider a normal compact K\"ahler klt space $X$ whose
canonical divisor $K_X$ is pseudo-effective, and give a dynamical criterion for
$X$ to be a $Q$-complex torus. We show that, if such $X$ admits an
int-amplified endomorphism, then $X$ is a $Q$-complex torus. As an application,
we prove that, if a simply connected compact K\"ahler (smooth) threefold admits
an int-amplified endomorphism, then it is (projective and) rationally
connected.
