  In this paper, we continue the study of Serrano's conjecture in low
dimensions. We focus on two special cases of the log version of Serrano's
conjecture: the ampleness conjecture and the log version of
Campana--Peternell's conjecture. In dimension 3, we prove that the ampleness
conjecture holds for non-canonical singularities; by the same method, we also
prove that the log canonical version of Campana--Peternell's conjecture holds
in dimension 3. In dimension 4, we improve the results on Campana--Peternell's
conjecture by excluding the case that the numerical dimension of the
anti-canonical divisor is 3. Specifically, we show that for a projective smooth
fourfold $X$, if $-K_X$ is strictly nef but not ample, then $\kappa(X, -K_X)=0$
and $\nu(X, -K_X)=2$; in this case, if we further assume that $X$ admits a Fano
contraction $X\to Y$ onto a surface $Y$ induced by some extremal ray, then
$\rho(X)=2$.
