Does the proxy have to insert a Record-Route header in every request so as to remain in path?
A. A proxy would remain in path if it had inserted a Record-Route header in the request that initiated the dialog. The only point where the proxy might move out of the path is when an endpoint reconstitutes a dialog after a failure(like a crash!). In such cases, a proxy should insert a Record-Route header in every request irrespective of it being part of the dialog. But this is optional i.e a proxy may wish to only insert a Record-Route in a request that originated the dialog.
