Oh, Conjury can heal deadly diseases - early Gridanian quests actually touched on that (a quest in Hyrstmill mentioned about the 'wasting sickness' that once decimated the hyur population of Gridania to the point it reached epidemic levels, but that now conjury spells exist that can treat and cure the disease - the context of the quest was that CNJs knowledgeable in that particular spell may not always be around so herbal treatments were still stored as a last resort).

As well, the spell Esuna (when it was still a CNJ spell) seemed to have the context of being a treatment for disease as well (after all, there is an actual status effect called 'disease' in the game). So CNJs (and by extension, WHMs) most definitely can heal illness and disease with their magic.