Conjecture on recursive uniformizers and eventual stability of truncated expansions

Let $p\geq 3$ be prime, let $n\geq 1$ and $m\geq 2$, and use the elements $\pi_p^{m,n}$, the fields $\mathbb K_p^{m,n}$, and the valuation $v_p$ from the construction. A series of polynomials means a family $\{\mathcal R_p^{m,n}(T)\}_{m\geq 2}$ in $\mathbb Z_{(p)}[T]$. For a power series $\mathcal R$, let $\operatorname{Trun}_{\leq u}(\mathcal R)$ denote its truncation to terms of degree at most $u$.

Recursive uniformizer and stability conjecture. For each prime $p\geq 3$ and integer $n\geq 1$, there exists a series of polynomials $\{\mathcal R_p^{m,n}(T)\}_{m\geq 2}$ such that, for every $m\geq 2$,

$$\pi_p^{m+1,n}=\left(\zeta_{p^{m+1}}-1\right)^{s_p^{m,n}}\left(\zeta_{p^{m+1}}-\mathcal R_p^{m,n}\left(\pi_p^{m,n}\right)\right),$$

where

$$s_p^{m,n}=\frac{1}{p^n}-p^m(p-1)v_p\left(\zeta_{p^{m+1}}-\mathcal R_p^{m,n}\left(\pi_p^{m,n}\right)\right)\in\mathbb Z,$$

is a uniformizer of $\mathbb K_p^{m+1,n}$. Moreover, for each fixed $n\geq 1$ and $m\geq 2$, there exist a non-negative integer $h_{m,n}$, power series $\mathcal W_0^{m,n}(T),\ldots,\mathcal W_{h_{m,n}}^{m,n}(T)$ in $\mathbb Q[[T]]$, and polynomials $u_{-1}^{m,n}(T)=-1,u_0^{m,n}(T),\ldots,u_{h_{m,n}}^{m,n}(T)$ in $\mathbb Z[T]$, with $-1<u_0^{m,n}(p)<\cdots<u_{h_{m,n}}^{m,n}(p)$ for every prime $p\geq 3$, such that

$$\mathcal R_p^{m,n}(T)=\sum_{i=0}^{h_{m,n}}T^{u_{i-1}^{m,n}(p)+1}\operatorname{Trun}_{\leq u_i^{m,n}(p)}\left(\mathcal W_i^{m,n}(T)\right).$$

If these power series and polynomials are chosen properly, the family is eventually stable: for sufficiently large $m$,

$$\mathcal R_p^{m,n}(T)=\mathcal R_p^{m+1,n}(T)=\mathcal R_p^{m,n+2}(T)=\cdots.$$

The conjecture is motivated by the difficulty of constructing uniformizers of $\mathbb K_p^{m,n}$ from the currently available truncated expansions of $\zeta_{p^n}$. The preceding theorem establishes the analogous construction for $\pi_p^{m,1}$, while the proposed recursive description and eventual stability for general $n$ remain conjectural.