Yes, seriously, this is getting out of hand. We are getting suggestions such as Kyiv (as if being Rus was basically being Roman) and England (when England did not even adopt the Roman law to a great extent compared to most of Europe) skipping most of the Romance language speaking countries that were basically Romans who assimilated the invaders like Goths or Franks even though regional identities eventually completely displaced the Roman identity, unlike England where the conquerors assimilated the conquered. We did not even get into other Byzantine "rump states" like Venice and Arborea that are just Byzantine provinces that became independent because of lack of control, so they were clearly Roman empires as well, right? Unlike almost all of the other examples Byzantines actually toke their "claims" to the legacy of Rome seriously, I mean much more seriously the almost everyone in the HRE, they even actually considered themselves the actual Romans. Constantinople clearly had the strongest claim to being the successor of the pre-1204 empire to the point that they were able to make the other successor states drop the title of the Roman emperor as mentioned by
@Aquila SPQR. Even on modern history maps Nicaea gets called Byzantine/Roman empire after 1261. I guess that after fending off those who "like the established historical terminology" we now need to face those who don't like it. Most of the HRE or even Aragon could be considered rump states of France i. e. Francia i. e. Frankish empire, but we don't get people questioning what is the real France.