CT GOP was derelict (what else is new) in not getting a top tier candidate (all of them were RINOS but still) into that 2006 race.
I think it was deliberate.
Something like Rell’s top aide is married to Lieberman staffer.
Had a top-tier, or even a second-tier, Republican been running, he or she would have had a real chance of winning a plurality in the general even in 2006, which obviously would change the equation drastically. Under such circumstances, Republicans should have supported the GOP nominee and not even given a second thought to Lieberman being a bit less bad than Lamont. But nobody with a decent chance to win ran, and then Schlesinger won the nomination, was found out to be a scumbag, and refused to drop out and be replaced with a viable candidate, thus making it clear that either Lieberman or Lamont would win the election.