Berger's invariant-elements conjecture for overconvergent substitutions

Let $K$ be a $p$-adic field, let $\mathcal{R}$ be the Robba ring with coefficients in $K$, and let $s$ be an overconvergent substitution of $\mathcal{R}$. The fraction field $\operatorname{Frac}\mathcal{R}$ consists of fractions of elements of $\mathcal{R}$.

Berger's conjecture.

$$(\operatorname{Frac}\mathcal{R})^{s=1}=K.$$

Thus the only elements of the fraction field fixed by the overconvergent substitution should be the constants in $K$. The source presents this as Berger's conjecture and states that its incompatibility with Kedlaya's conjecture is shown in the anticyclotomic setting; no resolution of the conjecture itself is given.