Let $U$ be a smooth complex variety and $X$ a compactification by a normal crossings divisor $D$. Let $E$ be a vector bundle on $U$ (i.e. locally free $\mathcal{O}_U$-module), together with a connection $\nabla: E \to E \otimes_{\mathcal{O}_U} \Omega^1_U$. Assume that:
(1) $\nabla$ is flat
(2) $(E, \nabla)$ has regular singularities, meaning that it can be extended to a connection $\nabla_X: E_X \to E_X \otimes \Omega^1_X(log D)$ with logarithmic poles.
Is $(E_X, \nabla_X)$ flat?