Irreducible common-zero conjecture for E-functions

Let $h_1$ and $h_2$ be irreducible $E$-functions, meaning non-units that cannot be written as products of two non-units.

Irreducible common-zero conjecture. If $h_1$ and $h_2$ have at least one common zero in $\mathbb C$, then

$$\nh_1=u h_2$$

for some unit $E$-function $u$.

This is a common-factor principle for irreducible $E$-functions and would imply that irreducible factors are determined, up to units, by their zeros. The source notes only the special case of linear factors associated with algebraic zeros; the general statement remains open.