Remark on equicharacteristic analogue of Hesselholts conjecture on cohomology of Witt vectors

Let $L/K$ be a finite Galois extension of complete discrete valued fields of characteristic $p$. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer $ngeq 0$, let $W_n(sO_L)$ denote the ring of Witt vectors of length $n$ with coefficients in $sO_L$. We show that the proabelian group ${H^1(G,W_n(sO_L))}_{nin N}$ is zero. This is an equicharacteristic analogue of Hesselholts conjecture.