  In this paper, we develop the theory of relative log convergent cohomology of
radius $\lambda$ ($0 < \lambda \leq 1$), which is a generalization of the
notion of relative log convergent cohomology in the previous paper. By
comparing this cohomology with relative log crystalline cohomology, relative
rigid cohomology and its variants and by using some technique of hypercovering,
we prove a version of Berthelot's conjecture on the overconvergence of relative
rigid cohomology for proper smooth families.
