One of the Black Ajah hunters took the opportunity to tell one of the others that she had a mustache, when they were messing around with the Oath Rod. There's your lie. Now you're going to claim she was actually Black, with zero evidence, because that is the entirety of your case against Egwene.

Pretty sure that was done when the Oaths were removed to test the application of their test. Wether or not a moustache was involved, I can't tell you. But that isn't the Gotcha Answer you were looking for. It was very direct that it was done without the Oaths and for a specific purpose.

Are you telling me Egwene had the Oaths removed for this conversation then has them put back on?