Roth-type approximation conjecture for values of E-functions

Let $f$ be an $E$-function and let $z_0\in\overline{\mathbb Q}$. For any $\varepsilon>0$, there exists $c>0$ such that for every $(p,q)\in\mathbb Z\times\mathbb N$ with $q\neq 0$, either $qf(z_0)-p=0$ or

$$\left|f(z_0)-\frac{p}{q}\right|>\frac{c}{q^{2+\varepsilon}}.$$

This is Roth's theorem when $f(z_0)$ is algebraic; for general values of E-functions, the conjecture predicts the optimal irrationality measure and remains open.