I haven't read through all the pages of this thread so apologies for that. I think that Brienne's alive and that she told Cat she'd kill Jaime. But I also think that she won't be able to do it and that something will drive her to kill Cat. Her story will echo Jaime's because she's going to realise that the person she swore herself too, Catelyn Stark, has become a monster, just like Jaime realised Mad King Aerys was a monster. And she'll kill Cat to save others, just as Jaime killed Aerys to save everyone in King's Landing. Jaime and Brienne are similar in so many ways, both thought to be kingslayers, both knights of the king's guard, both loyal to each other. It makes sense that it'll play out that way because then Brienne will finally understand what Jaime went through to break his oath the way he did, even though she reviled him for so long for being an oathbreaker.