Family member playing mind games - do you step in?

For reasons I don’t fully understand, my older sister (who does not live at home anymore) has had some kind of a grudge against our mother for a few years. She had a rocky relationship with both parents during her teenage years; for some reason she got past that with Dad and is close to him now, but not Mom.

The result is that she scrutinizes everything my mom says to her much more than anyone else, and she’s always finding reasons to get offended. Mom says she feels like she has to walk on eggshells around her, and she has noticed that she doesn’t treat other people the same way, but doesn’t understand why.

I don’t fully understand why myself, but I just feel bad for my poor mom in this situation. And I also feel partially responsible because I think she feels very resentful about the fact my mom treats me differently than she does my sister, due to the very different nature of our relationship.

Probably I should just stay out of it, but I wish I could encourage my sister to just talk to my mom about whatever is the matter, instead of playing mind games and sending passive-aggressive signals whenever they interact. Or I wish I could give my mom some pointers on how to change the way she talks to her, because I do recognize a few habits my mom has that seem to bother my sister.

Advice?