Invariants conjecture for overconvergent substitutions in the Robba ring

Let $K$ be the coefficient field, let $\mathcal{R}$ be the Robba ring, and let $\varphi$ be the substitution induced by $s(X)$. The substitution $\varphi$ is overconvergent if $s(X)$ lies in the ring of integers of $\mathcal{E}^\dagger$ and its reduction $\overline{s}(X)\in k((X))$ is nonzero and belongs to $X^2\cdot k[[X]]$.

Invariants conjecture. If $\varphi$ is overconvergent, then

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

The theorem proved in the paper establishes this equality for substitutions of finite height; the conjecture extends the expected description of invariant elements to all overconvergent substitutions.