Tyrion is friendly with several of the Starks, and while he doesn't fall in love with Arya, he does marry Sansa. Tyrion didn't kill Joffrey, but he was blamed for the king's murder.

After Joffrey ends his engagement to Sansa, Tywin orders Tyrion to marry her, even though he is significantly older. Neither Sansa nor Tyrion want this marriage, and so it's no surprise that the ...

She was Tyrion’s wife at the time, which would have made him look guilty as well. Olenna and the Tyrell faction may have wanted Sansa to fall from grace alongside Tyrion.