Splitting conjecture for the wild inertia extension of a local field

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, with maximal unramified extension $K^\mathrm{nr}$ and separable algebraic closure $K^s$. Let $P=\operatorname{Gal}(K^s/K^\mathrm{tr})$ be the wild inertia subgroup, and identify the tame quotient with $\widehat{\mathbb{Z}}(1)(\neg p)$. Splitting conjecture. The extension

$$1\to P\to \operatorname{Gal}(K^s/K^\mathrm{nr})\to \widehat{\mathbb{Z}}(1)(\neg p)\to 1$$

splits, so that

$$\operatorname{Gal}(K^s/K^\mathrm{nr})\cong P\rtimes\widehat{\mathbb{Z}}(1)(\neg p).$$

The claim describes the inertia group as a semidirect product of its pro-$p$ wild part and its prime-to-$p$ tame quotient. The source notes that the splitting would be immediate for finite kernel or quotient, but does not state a resolution in the general profinite case.