Ritt-type factorization conjecture for E-functions

Let $\mathcal E$ be the ring of $E$-functions. A unit is a non-vanishing $E$-function. An $E$-function is simple if it has the simple-function structure defined in the source, and its support is the associated one-dimensional $\mathbb Q$-vector space. An $E$-function is irreducible if it is a non-unit that cannot be factored into two non-units. A non-zero $E$-function is normalized when its Taylor expansion has first nonzero coefficient equal to $1$ and the following coefficient equal to $0$.

Ritt-type factorization conjecture. For every non-zero $E$-function $f$, there exist a unit $u\in\mathcal E^\times$, simple normalized $E$-functions $s_1,\ldots,s_p$ with pairwise distinct supports, and irreducible normalized $E$-functions $h_1,\ldots,h_n$, where $p,n\geq0$, such that

$$\nf=u s_1\cdots s_p h_1\cdots h_n.$$

Moreover, $u,s_1,\ldots,s_p,h_1,\ldots,h_n$ are unique. This would provide the conjectural analogue of Ritt's factorization theorem for $E$-functions and remains open.