  This is the second paper of two in a series under the same title ([CRX]);
both study the quantitative volume space form rigidity conjecture: a closed
$n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is
diffeomorphic to a $H$-space form if for every ball of definite size on $M$,
the lifting ball on the Riemannian universal covering space of the ball
achieves an almost maximal volume, provided the diameter of $M$ is bounded for
$H\ne 1$.
  In [CRX], we verified the conjecture for the case that $M$ or its Riemannian
universal covering space $\tilde M$ is not collapsed for $H=1$ or $H\ne 1$
respectively. In the present paper, we will verify this conjecture for the case
that Ricci curvature is also bounded above, while the above non-collapsing
condition is not required.
