When Gentiles convert to Judaism, they are "like a newborn".
If I understand correctly, there is no such thing as Niddah vis a vis non-Jews.
So, if I add those two together, I am inclined to assume that if a woman has just finished her period when she goes to the Mikvah for her conversion, not only is she Jewish, but she is in a state of purity from the very start, without counting 7 clean days, etc.
If that assumption is correct, what about if she were to go to Mikvah during her period? Would her conversion be valid, or would her ongoing (or, really, new) state of impurity invalidate the immersion and, by extension, the conversion?