The parent-child relationship of a page is not updated during a move

The ancien parent field is updated during creation process, and thanks to XWIKI-13106, the children are also updated during a rename of the parent page. But, when a page is moved, the parent field of the page itself is not updated according to its new parent.
Moreover, the NS Migrator App also update that parent field in addition to the normal rename process. Issue linked to parent field get notice very late in legacy application, since the parent field is properly maintained in all cases but this one. So I believe this is a bug that need to be fix, or there is no point to have manage all the other use cases.

Step to reproduce:
1) Create page A(.WebHome)
2) Create page A.B(.WebHome)
3) Create page A.B.C(.WebHome)
4) Create page A.B.C.D(.WebHome)
5) Move page A.B.C(.WebHome) to page A.C(.WebHome)
5) Export all these pages to XAR, and check their <parent> tags.