I feel it would be foolish for SE to make CON the white mage.
First: The lore and structure of THM guild fits them taking on the white mage role. The THM worship the twin gods of life and death. White mage easily fits in to the lore and SE is huge on lore.
Second: Why give more healing to a class that already has the better heals? Why not give it another class, THM, and make two classes able to heal a group? The jobs are said to open up new abilities, spells for us, to the classes.
Third: Making a THM into black mage is too huge a stretch no matter how you look at it. You would have to recreate the class and lore. SE is changing things, but I fail to see them going that far.
Just my line of thinking.