Local freeness conjecture for families of $(\varphi,\Gamma_K)$-modules

Let $K$ be the $p$-adic field, let $A$ be the coefficient algebra, and let $\mathcal{R}_A(\pi_K)$ denote the associated Robba ring. A $(\varphi,\Gamma_K)$-module $M$ over $\mathcal{R}_A(\pi_K)$ is equipped with commuting semilinear actions of $\varphi$ and $\Gamma_K$. For an admissible affinoid algebra $A_i$ over $A$, write $\mathcal{R}_{A_i}(\pi_K)$ for the corresponding base-changed Robba ring and $M\widehat\otimes_A A_i$ for the completed base change.

Local freeness conjecture. For every $(\varphi,\Gamma_K)$-module $M$ over $\mathcal{R}_A(\pi_K)$, there exists a finite admissible cover $\{\operatorname{Max}(A_i)\}_{i=1,\dots,n}$ of $\operatorname{Max}(A)$ such that each $M\widehat\otimes_A A_i$ is a finite free $\mathcal{R}_{A_i}(\pi_K)$-module.

This conjecture would give local freeness after a finite admissible base change, providing useful control of families of $(\varphi,\Gamma_K)$-modules. The source presents it as a technical conjecture and points to related questions in the cited literature; no resolution is supplied here.