2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161294In this paper, we develop the theory of relative log convergent cohomology of radius $λ$ ($0 < λ\leq 1$), 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.80 pages, minor errors and typos fixedNumber TheoryAlgebraic Geometry14F30text