diff --git a/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h b/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h
index e709b9f6ee9..69aa9ead435 100644
--- a/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h
@@ -844,7 +844,7 @@ public:
   Conforming_constrained_Delaunay_triangulation_3 convert_for_remeshing() && {
     auto& tr = cdt_impl;
     for(auto vh : tr.all_vertex_handles()) {
-      vh->sync();
+      vh->sync_vertex_type_with_dimension_and_index(tr);
     }
 
     for(auto ch : tr.all_cell_handles()) {
@@ -875,43 +875,6 @@ public:
   /// \endcond
   // end SKIP_IN_MANUAL for convert_for_remeshing
 
-  /// \cond SKIP_IN_MANUAL
-  Conforming_constrained_Delaunay_triangulation_3 prepare_for_tetrahedral_remeshing() &&
-  {
-    auto& tr = cdt_impl;
-    for(auto vh : tr.all_vertex_handles()) {
-      vh->reset_for_remeshing(tr);
-    }
-
-    for(auto ch : tr.all_cell_handles()) {
-      ch->set_subdomain_index(1);
-    }
-
-    std::stack<decltype(tr.infinite_cell())> stack;
-    stack.push(tr.infinite_cell());
-    while(!stack.empty()) {
-      auto ch = stack.top();
-      stack.pop();
-      ch->set_subdomain_index(0);
-      for(int i = 0; i < 4; ++i) {
-        if(ch->is_facet_on_surface(i))
-          continue;
-        auto n = ch->neighbor(i);
-        if(n->subdomain_index() == 1) {
-          stack.push(n);
-        }
-      }
-    }
-    Conforming_constrained_Delaunay_triangulation_3 result{std::move(*this)};
-    // static_assert(CGAL::is_nothrow_movable_v<Conforming_constrained_Delaunay_triangulation_3>);
-    static_assert(
-        std::is_same_v<std::remove_reference_t<decltype(*this)>, Conforming_constrained_Delaunay_triangulation_3>);
-    *this = Conforming_constrained_Delaunay_triangulation_3{};
-    return result;
-  }
-  /// \endcond
-  // end SKIP_IN_MANUAL for convert_for_remeshing
-
   /**
    * A bidirectional iterator for visiting all constrained facets of the triangulation.
    * The value type of this iterator is `Triangulation::Facet`.
diff --git a/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_vertex_base_3.h b/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_vertex_base_3.h
index 8ee2371d566..721bee0c432 100644
--- a/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_vertex_base_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_vertex_base_3.h
@@ -76,39 +76,11 @@ public:
   std::size_t cached_number_of_incident_facets() const { return number_of_incident_facets_; }
   std::size_t cached_number_of_components() const { return number_of_components_; }
 
-  void sync() {
-    switch(ccdt_3_data().vertex_type()) {
-      case CDT_3_vertex_type::FREE:
-        set_dimension(3);
-        set_index(0);
-        break;
-      case CDT_3_vertex_type::CORNER:
-        set_dimension(0);
-        set_index(0);
-        break;
-      case CDT_3_vertex_type::STEINER_ON_EDGE:
-        set_dimension(1);
-        set_index(0);
-        break;
-      case CDT_3_vertex_type::STEINER_IN_FACE:
-        set_dimension(2);
-        set_index(ccdt_3_data().face_index());
-        break;
-      default:
-        CGAL_error();
-        break;
-    }
-  }
-
   template<typename Tr>
-  void reset_for_remeshing(const Tr& tr)
+  void sync_vertex_type_with_dimension_and_index(const Tr& tr)
   {
     switch(ccdt_3_data().vertex_type())
     {
-    case CDT_3_vertex_type::FREE:
-      set_dimension(3);
-      set_index(0);
-      break;
     case CDT_3_vertex_type::CORNER:
       set_dimension(0);
       set_index(0);
@@ -121,6 +93,10 @@ public:
       set_dimension(2);
       set_index(ccdt_3_data().face_index());
       break;
+    case CDT_3_vertex_type::FREE:
+      set_dimension(3);
+      set_index(1);
+      break;
     default:
       CGAL_error();
       break;
diff --git a/Constrained_triangulation_3/include/CGAL/make_conforming_constrained_Delaunay_triangulation_3.h b/Constrained_triangulation_3/include/CGAL/make_conforming_constrained_Delaunay_triangulation_3.h
index 0f5a265a0e2..6de883acd4d 100644
--- a/Constrained_triangulation_3/include/CGAL/make_conforming_constrained_Delaunay_triangulation_3.h
+++ b/Constrained_triangulation_3/include/CGAL/make_conforming_constrained_Delaunay_triangulation_3.h
@@ -123,7 +123,7 @@ auto make_conforming_constrained_Delaunay_triangulation_3(const PolygonMesh &mes
   using CDT = typename CGAL::Default::Get<Triangulation,
                                           Conforming_constrained_Delaunay_triangulation_3<Mesh_geom_traits>>::type;
   CDT cdt(mesh, np);
-  auto remeshing_cdt{std::move(cdt).prepare_for_tetrahedral_remeshing()};
+  auto remeshing_cdt{std::move(cdt).convert_for_remeshing()};
   static_assert(std::is_same_v<decltype(remeshing_cdt), CDT>);
   return remeshing_cdt;
 }