diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
index 4b41ece8f23..403cfaca69b 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
@@ -394,22 +394,23 @@ public:
     return cells_with_too_big_orthoball.empty();
   }
 
+  // returns 'true/false' depending on whether the cover would (or has, if 'abort_if_cover_change'
+  // is set to 'false') change.
   bool update_cover_data_during_management (Cell_handle new_ch,
                                             const std::vector<Cell_handle>& new_cells,
                                             const bool abort_if_cover_change)
   {
-    bool result = false;
-
     if(compare_orthsphere_radius_to_threshold(new_ch, orthosphere_radius_threshold) != CGAL::SMALLER)
     {
       if(is_1_cover())
       {
-        if(abort_if_cover_change)
-          return true;
-
+        // Whether we are changing the cover or simply aborting, we need to get rid of the new cells
         tds().delete_cells(new_cells.begin(), new_cells.end());
-        this->convert_to_27_sheeted_covering();
-        result = true;
+
+        if(!abort_if_cover_change)
+          this->convert_to_27_sheeted_covering();
+
+        return true;
       }
       else
       {
@@ -417,7 +418,7 @@ public:
       }
     }
 
-    return result;
+    return false;
   }
 
   virtual void update_cover_data_after_converting_to_27_sheeted_covering ()
diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
index b7223efef7e..9103e88afb2 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_triangulation_3.h
@@ -3247,14 +3247,8 @@ periodic_remove(Vertex_handle v, PointRemover& remover, CoverManager& cover_mana
     if(cover_manager.update_cover_data_during_management(new_ch, new_cells,
                                                          abort_if_cover_change))
     {
-      // If we have switched to 27-sheet, 'new_ch' has already been cleaned up,
-      // but we must clean it if we have aborted a 27-sheet conversion.
-      if(is_1_cover() && abort_if_cover_change)
-        _tds.delete_cell(new_ch);
-
       CGAL_triangulation_expensive_postcondition(_tds.is_valid());
-
-      return false; // removing would cause / caused a cover change
+      return false; // removing would cause / has caused a cover change
     }
 
     // The neighboring relation needs to be stored temporarily in