Jossen's factorization conjecture for E-functions

Let $\mathcal E$ be the ring of $E$-functions. A unit is a non-vanishing $E$-function, equivalently a function $\lambda e^{\alpha x}$ with $\lambda,\alpha\in\overline{\mathbb Q}$ and $\lambda\neq0$. An $E$-function is irreducible if it is a non-unit that cannot be expressed as a product of two non-units. A simple $E$-function has the simple-function structure defined in the source, with a one-dimensional rational support.

Jossen's factorization conjecture. Every non-zero $E$-function $f$ can be written as a finite product of powers of $E$-functions with simple zeros, with distinct factors having no common zero; explicitly, there is a finite $J\subset\mathbb N^*$ such that

$$\nf=\prod_{j\in J}g_j(x)^j,$$

where each $g_j$ is an $E$-function whose zeros have multiplicity $1$, and $g_j$ and $g_{j'}$ have no common zero for $j\neq j'$. The representation is unique up to multiplication of the $g_j$ by units.

This is the precise factorization form of the preceding multiplicity claim. It is intended to describe the structure of zeros and remains open.