It's not your responsibility to fix the relationship your mother has with your sisters. Your sisters have decided that their relationship with their mother tends to lead to them feeling upset and your mother expects them to give into HER needs. From what you have shared, your mother expects you and your sisters to bend and give in while she stands her ground even when she is wrong. Your mother has to learn to accept when she is wrong and that her children deserve and have a right to have their own opinions on things.

Your mother in three years has not admitted she was wrong and that her needing to continue to insist even when she was wrong is why her daughters don't want to be around her. Your mother doesn't care to be responsible for her own "wrongs". This is something YOU cannot fix. I am sure you want your mother and sisters to find a way to get along better. Unfortunately, your mother is the one that has created this problem in that she is not willing to admit her own wrongs, not even when others show her facts proving she is wrong. Plus, your mother emotionally manipulates you by expecting you to fix HER wrongs. This is just another way your mother is STILL after three years now refusing that she was wrong in that arguement and how poorly she treated your sisters driving home and when they were to sleep at your mother's house. The only recourse you have is to say to your mother, "Mom, your unwillingness all this time to admit you were wrong even when your daughters proved you wrong is why they don't want to connect with you. Your expecting me to fix YOUR unwillingness to admit when you are wrong is not something I can do. It's something YOU have to do and only YOU."