I think if she had been proven right it would've become a big cultural thing. They would've made a big-budget movie, tons of romanticized biographies about her would've been released, etc. Not saying that hasn't happened anyway, but if we knew she was for real it would all be that times a million.
Now that we all know she was a fake hopefully in a century people will remember the real Anastasia and her family much more than AA, who will fall into obscurity like most royal pretenders throughout history did.