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Let $X$ be a quasi-projective variety over a field $k$ of characteristic zero. A good compactification of $X$ means a projective variety $\overline{X}$ containing $X$ as the complement of a simple normal crossings divisor $D$.

Let $E$ be a locally free sheaf over $X$, equipped with a flat connection $\nabla: E \to E \otimes \Omega^1_X$.

Definition: We say that $(E, \nabla)$ has regular singularities at infinity if, for all good compactifications $\overline{X}$ of $X$, there exists a locally free sheaf $\overline{E}$ on $\overline{X}$ and a logarithmic connection $\overline{\nabla}: \overline{E} \to \overline{E} \otimes \Omega^1_{\overline{X}}(\log D)$ extending $(\nabla, D)$.

In "The regularity theorem in algebraic geometry", Katz explains that it suffices to check that, for (only) one given smooth compactification, there exists a coherent sheaf $\overline{E}$ with a logarithmic connection like above. Notice that, contrary to the case of connections "without poles" this does not imply that $\overline{E}$ is locally free.

However, sometimes I have seen the definition of regular singular connections with only one compactification and $\overline{E}$ locally free. Is this definition equivalent to the previous one? If this is not the case, what would be a counterexample?

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