The division conjecture for arithmetic Gevrey functions

Let $\mathfrak{f}(z)$ be an arithmetic Gevrey function and let $\theta\in(-\pi/2,\pi/2)$ satisfy $\mathfrak{f}_\theta(1)=0$. Division conjecture for arithmetic Gevrey functions. Then

$$\frac{\mathfrak{f}(z)}{z-1}$$

is an arithmetic Gevrey function. The conjecture is intended as an analogue of the corresponding division property for $E$-functions; the source notes related discussion by André and an earlier formulation by Ferguson in the rational-coefficient, $\theta=0$ case.