Convergence conjecture for substitutions in the Robba ring

Let $\mathcal{R}$ be the Robba ring associated with a finite-height substitution $\varphi$ induced by $s(X)$, and let $\rho(s)$ be the largest norm of a zero of $s$ in the open unit disk. For $r<1$, write $\mathcal{R}^{[r;1[}$ for the subring of elements convergent on the annulus with radii $r\leqslant |z|<1$, and let $\mathcal{R}^+$ denote the positive-power subring.

Convergence conjecture. If $h\in\mathcal{R}$ is such that $\varphi(h)$ is convergent on the annulus

$$\{z,\,\rho(s)\leqslant |z|<1\},$$

then $h\in\mathcal{R}^+$.

The paper proves this conjecture in the cyclotomic case $s(X)=(1+X)^p-1$. It predicts that convergence of the substituted element on the largest forced annulus already implies that the original element has no negative powers.