I'm a Liverpool fan over here in the UK as well as a Red Sox follower.
I have to say that there has been no sign of FSG showing any less interest in the Red Sox from where we are. Infact, they flew over for Opening Day at the weekend missing our FA Cup Semi Final at Wembley against our City rivals Everton. They also missed the Hillsborough memorial service the following day which is an improtant day in the LFC calander. Not that any of us hold any resentment for missing either, we understand that Opening day on Fenways 100yr anniversary is also a big deal.
There have been problems at LFC, but they took over a total wreck of a football club. Slowly they have built us back up again, however on the field there is much to do. The English Premier League however is a massive cash cow. More so than any other sport in the world. The Champions League is a money generator and I think they will be putting a lot of energy into getting us back into that once more. Add to that Anfield either needs redeveloping or a new ground needs building. Once they are achieved, I think you will see a bit more balance between the two of us.
I can't see them selling the Red Sox though. It, along with LFC, is a flagship brand of their portfolio.