Jossen's conjecture on division and common zeros of E-functions

An $E$-function is a power series $f(x)=\sum_{n=0}^{\infty}a_nx^n/n!\in\overline{\mathbb Q}[[x]]$ satisfying the three defining growth, integrality, and differential-equation conditions. A unit is an $E$-function of the form $\alpha e^{\beta x}$ with $\alpha,\beta\in\overline{\mathbb Q}$ and $\alpha\neq0$.

Jossen's conjecture. (i) If $f$ and $g$ are two $E$-functions such that $f/g$ is entire, then $f/g$ is an $E$-function. (ii) If two $E$-functions $f$ and $g$ share at least one common root, then there exists a non-unit $E$-function $h$ such that $f/h$ and $g/h$ are $E$-functions.

Part (i) is known for exponential polynomials over $\overline{\mathbb Q}$ and when $g$ is a polynomial; part (ii) is known when the common root is algebraic. The general conjecture remains open.