Let's Talk Politics?

The Andromeda Initiative is by all means, an independent civilian endeavor, like a private corporation trying to settle a new world. But these are still people of their home Governments, and hopefully if all goes well enough, the Initiative is going to turn into a major power in the Heleus Cluster, complete with bustling cities on Eos, Kadara, and Elaadan etc, as well as the Nexus Station, it's future projected 2M+ inhabitants, and an assumed future fleet of warships to defend it all.

Now, suppose, one day in the future they manage to successfully establish a means of feasible communication across Dark Space back to whatever civilizations have arisen from the Reaper War. What are the odds you think these new Governments will want to try and extend influence or even dominion over the Initiative? It's not like Earth can send an Armada over to Andromeda to enforce the new Human colony worlds under the banner of the Systems Alliance, (or whatever exists now). If there was a means of making the travel between Galaxies less lengthy, then the desire for control would grow as Colonies in Andromeda would be a major boost in power for whoever back home.

Perhaps in those 630 years Earth and the others had begun to deploy their own Mass Relay network out into Dark Space in a long term plan to bridge the gap and create a corridor, similar to how it was done in the show Stargate SG1, where they took and placed a series of Stargates between the two Galaxies as a bridge. It's not unlikely that after the Reaper War the home Races would've put extensive work into studying the Mass Relays, even if they were busted up after the Destroy Ending of 3, and Matriarch Aethyta even mentioned she tried to suggest the Asari start building their own Relays.

If the conditions were met, would you think the Governments back home would try and establish rule over the Andromeda Colonies?

Or perhaps they would recognize the Initiative as a Sovereign State?