Bessel common-zero conjecture for E-functions

Let $\alpha\in\mathbb Q\setminus(\{\pm1/2\}\cup(-\mathbb N^*))$, and let $f$ be an $E$-function. The Bessel function $J_\alpha$ is understood through its normalized $E$-function $\Gamma(\alpha+1)x^{-\alpha}J_\alpha(x)$.

Bessel common-zero conjecture. If $f$ and $J_\alpha$ share a common zero in $\mathbb C^*$, then

$$\nf(x)=g(x)\Gamma(\alpha+1)x^{-\alpha}J_\alpha(x)$$

for some $E$-function $g$.

This is a proposed extension of the common-factor principle from exponential polynomials to general $E$-functions. Its general case remains open.