Let $K$ be a local field of characteristic $p$, and $K^{sep}$ be its separable closure. Let $F_q : W(\mathcal K) \longrightarrow W(\mathcal K)$ be the homomorphism of Witt vectors $$ (w_0, w_1, \ldots ) \longmapsto (w_0^q, w_1^q,\ldots) $$ given by $F^{N_0},$ where $q = p^{N_0}$, F being the Frobenius map, $\mathcal K$ being any characteristic $p$ field. It can be shown that for each $A \in W(K)$, there exists $\overline W \in W(K^{sep})$ such that $$F_q(\overline W) - \overline W = A. \tag{$*$}$$ This can be seen in Proposition 8 of Dongryul Kim's blog post. The sketch is that for $\overline W = (w_0, w_1,\ldots)$ we get a condition of the form $$ w_i^q - w_i = f_i(w_0, w_1,\ldots , w_{i-1} ; a_i)$$ which is separable over $K(w_0, w_1,\ldots, w_{i-1})$ and hence over $K$.
We can define an action of $\text{Gal}(K^{sep}/K)$ on $W(K^{sep})$, given by $$ \sigma: (w_0, w_1, \ldots) \longmapsto (\sigma(w_0), \sigma(w_1), \ldots), $$ which is justified in the same blog post mentioned above. I have been told it is possible to show that $$ \sigma(\overline W) - \overline W \in W(\mathbb F_q), $$ where $\overline W$ is a solution to $(*)$ above, and I would like your help in this endeavour.
Writing $V = \sigma(\overline W) - \overline W$ as $(v_0, v_1,\ldots)$, I think we need to show that $v_n \in \mathbb F_q$ for all $n$.
For $n=0$ this is straightforward, as $$ v_0 = \sigma(w_0) - w_0,$$ which lies in $\mathbb F_q$ as $\sigma : w_0 \longmapsto w_0 + i$ for some $i \in \mathbb F_q$.
For $n>0$ is where I haven't been able to solve it. Consider the case $n=1$, we have $$ v_1 = \sigma(w_1) - w_1 + \frac{ \sigma(w_0)^p - w_0^p - (\sigma(w_0) - w_0)^p }{p} \tag{$\dagger$} $$ It is not clear to me that the RHS lies in $\mathbb F_q$, since we have the term $\sigma(w_0)^p - w_0^p$, for which the image in $\mathbb F_q$ is not clear to me at all.
I did try making an explicit computation in the case where $p=3$, but it was not particularly fruitful. The expression I obtained for the second term of $(\dagger)$ was $$w_0^2 i + w_0i^2,$$ where $i = \sigma(w_0) - w_0$, which doesn't seem to ring any bells to me.
Any advice is appreciated, thank you!