Regularity of coordinate sequences on logarithmic Wach modules

Establish whether, for every index $1\leqslant i\leqslant d$, the sequences $\{p,\mu,[x_i^{\flat}]\}$ and $\{\mu,p,[x_i^{\flat}]\}$ are regular on every Wach module over $A_R^+$, where $A_R^+$ is the logarithmic Wach-module coefficient ring and $[x_i^{\flat}]$ denotes the Teichmüller lift of the compatible system of $p$-power roots of $x_i$.

Background

The paper proves in Lemma 4.17 that the sequences {p,μ,[xi♭]}\{p,\mu,[x_i^{\flat}]\} and {μ,p,[xi♭]}\{\mu,p,[x_i^{\flat}]\} are regular on the Wach-module-like object AR+(T)A_R^+(T) attached to a finite free representation. Proposition 4.18 provides comparison inclusions relating an arbitrary effective Wach module NN to this representation-theoretic object. The authors then explicitly ask whether the same regularity property holds for every Wach module over AR+A_R^+, rather than only for those arising from representations through the constructed comparison.

References

For each $1 \leqslant i \leqslant d$, are the sequences ${p, \mu, [x_i{\flat}]}$ and ${\mu, p, [x_i{\flat}]}$ regular on a Wach module $N$ over $+$?

— Log-crystalline representations and $(\varphi, Γ)$-modules  (2609.11266 - Abhinandan et al., 10 Sep 2026) in Question immediately following Proposition 4.18 in Section 4, “Wach modules”