The division conjecture for mixed arithmetic Gevrey functions

A mixed function is a formal series

$$\Psi(z)=F(z)+\mathfrak{f}(1/z),$$

where $F$ is an $E$-function and $\mathfrak{f}$ is an arithmetic Gevrey function. For a non-anti-Stokes direction $\theta$, write $\Psi_\theta$ for the corresponding sectorial function. Division conjecture for mixed functions. If $\theta\in(-\pi/2,\pi/2)$ and $\Psi_\theta(1)=0$, then

$$\frac{\Psi(z)}{z-1}$$

is a mixed function. This conjecture is stated as implying both the division conjecture for arithmetic Gevrey functions and the earlier conjecture in the paper; the source does not report a resolution.