The fibrations constructed in Theorem 6.19 have smooth Lagrangian sections and the action coordinates extend continuously to the whole base. Since $\mathcal{F}|_{U}$ also has a Lagrangian section (cf. Remarks 7.7) and the action coordinates extend continuously to the whole base, the statement easily follows from the results on stitched fibrations such as the existence of a normal form. The latter is found extending the maps $f^{+}$ and $f^{-}$ beyond all connected components of the seam and then using the Lagrangian section to normalize with the period map.

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