Any demand of being officially made "Third Rome" is uncanonical.
That being said, there is no doubt that the world's largest Orthodox Church is treated as a peripheral patriarchate by the EP.
Orthodoxy seems to be doing much better when it stays clear from ecumenism.
I can't see what the largest see should be batting 5th in the lineup.
Frankly I do not see why Russia (and when Filofey said Third Rome he wasn't reffering to the city of Moscow but the entire bounds of the Russian church) should not be at least 3rd among equals. Why should Sees which don't even still hold a calcuable number of Christians in their city of note be first in processions. Why should they be allowed to call councils when only a tiny fraction of the Orthodox church is under their Patriarch, but those with the vast majority of the Orthodox under them be unable to, and have to ignore their meddling in their jurisdictions?