Galois cohomology of Witt vectors of algebraic integers

Let L/K be finite Galois extension of complete discrete valuation fields of mixed characteristic with Galois group G and suppose that the induced extension of residue fields is separable. It is well-known that the group H1(G,OL) is zero if and only if the extension L/K is tamely ramified. We show, however, that the pro-abelian group H1(G,W.(OL)) is zero also for many wildly ramified extensions. Here W.(OL) is the pro-ring of Witt vectors in OL. We conjecture that this pro-abelian group is zero for all Galois extensions L/K with separable residue extension.