  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 $n\geq 0$, 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 ${H^1(G,W_n(\sO_L))}_{n\in \N}$ is zero. This is an
equicharacteristic analogue of Hesselholt's conjecture.
