The challenge that requires to complete a solo mission with an incarnon weapon equipped in every slot - DOESN'T COUNT if you are playing with TITANIA, probably because the game counts the EXALTED weapons as non-incarnon and therefore you aren't equipped with full set of incarnon weapons. Haven't tried it on other Warframes except Wukong, with whom I completed the challenge.
Thought of reporting it as another bug, so it would be great if this is fixed to avoid any similar situations in the future.