Might not be dead, might just be estranged. Nao phrased her going to see her family as being "awkward", it could be they didn't support her desire to go professional and she left home once she was old enough, leading to her becoming estranged from them, and in turn leaving Sugu back home. Sugu is likely a Fantome Fighter, and her desire from Fantasma could be that she doesn't want Nao to leave her behind this time.