There is the sadly cynical view, that I'm sure has more than a grain of truth, that in asking the big 6 to accept each others certifications you are asking turkeys to vote for Christmas.
For example, I don't see many signs of willingness in EASA to do that wholeheartedly, although quite a lot of harmonisation has take place and more is on the way; harmonisation does not necessarily mean accepting certifications.
I don't know much about the others but I suspect that the FAA would pull up the drawbridge when it comes to endorsing - without question - another Administration's certification.
Maybe the process will be gradual; start with small components, get that bedded in with mutual trust developing, then big components, assemblies, engines, whole aircraft.