Let $\mathcal{O}$ be the ring of integers in a $p$-adic local field, totally ramified over $\mathbb{Q}_p$. We fix a uniformizer $\pi$ and form the ring of relative Witt vectors $W_{\mathcal{O}}(\mathcal{O})$ with respect to $\pi$. See [Fargues, The Curve, p. 4] for a definition of $W_{\mathcal{O}}$. There is a natural map $$W(\mathcal{O})\longrightarrow W_{\mathcal{O}}(\mathcal{O}),$$ where $W(\mathcal{O})$ denotes the usual Witt vectors of $\mathcal{O}$. The map is determined by the property that it commutes with forming the ghost components. Let us consider its $\mathcal{O}$-linear extension $$\mathcal{O}\otimes_{\mathbb{Z}_p}W(\mathcal{O})\longrightarrow W_{\mathcal{O}}(\mathcal{O}).$$
Let $V_\pi$ denote the Verschiebung on $W_{\mathcal{O}}(\mathcal{O})$. It is known that there exists a unit $\varepsilon$ such that $$V_\pi(\varepsilon) = \pi - [\pi].$$ My Question is now as follows: Does this unit come from the ring $\mathcal{O}\otimes_{\mathbb{Z}_p}W(\mathcal{O})$?
Equivalently: The ghost of $\varepsilon$ is $(1-\pi^{p-1},1-\pi^{p^2-1},\ldots,1-\pi^{p^n-1},\ldots)\in \mathcal{O}^{\mathbb{N}}$. So the question is, whether $(\pi^{p-1},\pi^{p^2-1},\ldots)$ lies in the image of the $\mathcal{O}$-linear extension of the ghost map $$\mathcal{O}\otimes_{\mathbb{Z}_p} W(\mathcal{O})\longrightarrow \mathcal{O}^{\mathbb{N}}.$$
