I heard years ago from someone I trust completely that he observed Rav Moshe hold the door for women.
How sad is it that we lack common sense sometimes and replace long complicated halachic analysis for common sense. How sad is it that we live in a world where being a decent human has been replaced with “salad and halacha” “holding the door open for women and halacha” “pokemon and halacha” “riding the bus and halacha” “taking your kids to the park and halacha”. you get the point.