On the singularities of differential equations satisfied by $E$-functions

Let $\xi$ be a value, at an algebraic point, of a Siegel $E$-function. As a special case of a very general interpolation result, we prove that there exists an $E$-function $f$ such that $f(1)=\xi$, and such that 1 is not a singularity of the minimal differential equation satisfied by $f$. We prove that the same property does not hold at the point $0$, when $\xi$ is the value at a non-zero algebraic number of the Bessel function. This answers an analogue of a question asked by Yves Andr{\'e} for $G$-functions.