I think this will be handled in a similar way to Birgitte. She remembered all of her previous lives, in their entirety, until she was ripped out of TAR and put into reality again. Because it is physically impossible to hold such a detailed account of, say, 1000 previous lives, she has slowly, or not so slowly, started losing the knowledge. Soon, she should only be able to remember a couple of her previous existences...which is sad to think about, but a necessity for the pattern to remain intact.

I think Rand will have a similar experience; if he does remember all of them, that is.