Is there any way to get CGAL's polyline simplification to work for inner/shared boundaries?

I've been trying to carry out line simplification on polygons belonging to maps with the help of this CGAL guide, e.g. South Korea.

This is a screenshot of South Korea after line simplification with CGAL.

I carried out the line simplification by adding each polygon to CGAL::Constrained_triangulation_plus_2<CDT> ct and then running CGAL::Polyline_simplification_2::simplify(ct, Cost(), Stop(0.5));.

The outer boundaries will get simplified but the inner/shared boundaries (between provinces) will not. Is there any way to get the inner/shared boundaries simplified too?

I've also attempted to simplify each polygon individually and then combining them back together to form the whole country. The issue, however, is that each polygon would get simplified on its own, and so if Polygon1 gets simplified, there's no guarantee that the shared vertices of the adjacent Polygon2 will relocate to the same coordinates as Polygon1's vertices, and so intersections occur when you put them back together, as will happen with this South Korea map.

Thank you.