True or false, if the Cyan is printing fine, the problem cannot be the black / cyan print-head?
False. Although those colors print from the same head, they have completely separate channels of ink and electrical signals. Each color also has different pigments that affect the head differently. Each color is used in different proportions as well which means that the black could have more fires than the cyan.
Usually when one channel of a head is bad and the other is good, the first thing I look at is the dampers and manifold area. If the damper is old (more than a year or two) or there is air in the ink line, change it. Take the damper off and check the manifold for cracks or damage. If everything looks good, move on to the cap top. If it hasn't been changed in more than a year or has even the slightest damage, change it. After that, make sure your pump tubes are clear and not clogged (probably not the issue but worth checking).
One other thing to check is the bottom of the head. If there is a lot of ink build up, it will cause the head to not seal with the cap top and won't clean properly. If all else fails, try to flush the head with cleaning solution and a syringe. This is a last resort option as it can do more damage to the head if you are not careful. If that doesn't help, unfortunately, you will need a new head.