Headquarters is in the drift. Lots of churches spread all over.
One of the biggest ones seem to be on Aballon (but they also have churches on Absalom Station and Akiton and the Diaspora and pretty much everywhere that tech-loving 'people' or AI live).
My head-canon is that there is a shrine of some sort in the ducts/guts of Absalom as close to the Starstone as his Goblin/Ysoki/AI followers could get without raising an alarm.
its stated in the write up of Absalom that you cant even get to the Starstone currently. The whole area is tightly restricted but even the stations own engineers cant get in. You can only get to the outer containment area. The actual room housing the Stone is sealed off and no one can currently get to it. Though some swear to seeing mysterious entities moving about inside the chamber through the viewing ports at times.
|Egil, priest of Triune|
Given the Starstone predated Triune by a hell of a long time, I can't really see Triune's church having any special access. Interest? Maybe, though honestly, every deity probably has a lot of interest in the Starstone. It is, after all, pretty much the most powerful artifact in existence, empowered by multiple divine sacrifices and able to flat out grant godhood in a way even the deities can't.
( No, really, think about it: deities created by the Starstone pretty much zoom straight to "major god" status. Demigods raised up by existing deities have nowhere near the apparent heft or profile, which could simply be politics and subordination. Or it could be divine bequest starts you at God Level 1, while the Starstone Test starts you at God Level 10+. )