Consider the \(a\)-th étale cohomology module of a proper smooth scheme over the ring \(O\) of Witt vectors with coefficients in a perfect field of characteristic \(p>0\) and consider for any real number \(v\geq 0\) the ramification subgroup of the absolute Galois group of a quotient field of \(O\). The author proves that under a certain inequality condition about \(p\), \(a\), \(v\) and \(N\) the ramification subgroups act trivially on any subfactor in the cohomology module, which is annihilated by \(p^ N\). This theorem was stated as a conjecture by \textit{J.-M. Fontaine} [Invent. Math. 81, 515-538 (1985; Zbl 0612.14043)]. Some special cases were already proved by him and by the author. The results are based on the Fontaine-Messing theorem on the connection between the cohomological and the crystalline representations of the Galois group.