The simple answer is that no one knows for sure.

It's generally not good PR for either of them to openly dislike each other. Neither of them have said that they are friends either. On Music Station specials, you can generally tell if a singer is a fan of another singer if they sing along with a video clip being played or if their body language is positive. I have not noticed this in the past between Ayumi and Namie. On the other hand, I haven't seen them react negatively to each other when they stood next to each other in the past. It seems like a pretty neutral relationship but there is the expected tension due to being in the same line of work.