Quantitative volume space form rigidity under lower Ricci curvature bound II

This is the second paper of two in a series under the same title; 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 an 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 the first paper, we verified the conjecture for the case that the Riemannian universal covering space \tilde M is not collapsed. 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 on \tilde M is not required.