Coparenting advice needed!!!

So my fiance and I have been together since before his daughter was born. She just turned 4 last month. I'm a hairstylist and her mother has asked (demanded**) that I never cut her hair which is fine, that's her choice and it honestly doesn't matter to me.

Well we had my step daughter for her birthday(December 16th) I did two braids on the top of her head. She went home and told her mother and her grandparents that I cut her hair. We got a scathing message from her mother demanding to know why I was allowed to cut her hair and why wasn't she asked before hand. My fiance explained that I only braided it and it was never cut. Issue was let go. We did ask her about it the next time we had her. My fiance asked her why she said I cut her hair and my stepdaughter was honestly convinced that I cut it. "Because she did."

Today, she posts on facebook that my stepdaughter came up to her and told her, "we've been lying. She really did cut my hair." And now her mother is accusing us of telling her to lie to her. I'm honestly just baffled as to why my stepdaughter is lying and why she lied in the first place. I don't know if there is some sort coersion going on when she goes back to her mother. It's already an incredibly difficult coparenting situation and things like this makes it so much harder. I just don't know what to think. Should we ask her about it again? Her mother won't believe us and will try to use it against us in the future. It makes me feel like i should be on eggshells around a 4 year old.