Remark on equicharacteristic analogue of Hesselholt's conjecture on cohomology of Witt vectors

Let $L\mathrm{/}K$ be a finite Galois extension of complete discrete valued fields of characteristic $p$. Assume that the induced residue field extension $k_L\mathrm{/}{k}_K$ is separable. For an integer $n\ge 0$, let $W_n({\text{sO}}_L)$ denote the ring of Witt vectors of length $n$ with coefficients in ${\text{sO}}_L$. We show that the proabelian group ${H^1(G,W_n({\text{sO}}_L))}_{n\in \mathbb{N}}$ is zero. This is an equicharacteristic analogue of Hesselholt’s conjecture.