Wild inertia abelianization conjecture for local fields

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, let $\overline{k}$ be an algebraic closure of its residue field, and let $P=\operatorname{Gal}(K^s/K^\mathrm{tr})$ be the wild inertia subgroup. Define

$$\mathbb{W}(\overline{k})_0^\times=\varprojlim_{k'\subset\overline{k}} W(k')_0^\times,$$

where the limit runs over finite extensions $k'$ of the residue field and the transition maps are norms. Wild inertia abelianization conjecture. The topological abelianization of $P$ is naturally isomorphic to

$$\mathbb{W}(\overline{k})_0^\times.$$

This would identify the abelianized wild inertia with the completed group of principal Witt-vector units, equivalently with the corresponding additive Witt-vector object via the logarithm. The source presents this as part of a proposed description of the absolute inertia group and gives no resolution.