REF: Simplify StringDtype check in write_multi_index (GH#63414)

Author: zacharym-collins

pandas/io/pytables.py

@@ -3150,7 +3150,7 @@ def write_multi_index(self, key: str, index: MultiIndex) -> None:
             if isinstance(lev.dtype, ExtensionDtype):
                 # GH 63412
                 if isinstance(lev.dtype, StringDtype):
-                    lev = lev.astype(object)
+                    pass
                 else:
                     raise NotImplementedError(
                         "Saving a MultiIndex with an extension dtype is not supported."