Ant exclusion
Sooty mold is not a fungus to go after directly. The black film grows on honeydew, the sticky waste of aphids, soft scale, mealybugs, or whitefly, so the fix starts with those insects, and ants are what keeps them going: the ants harvest the honeydew and chase off the predators and parasites that would eat the insects producing it. Find the ant trail on the trunk and shut it down. Wrap the trunk with heavy paper, duct tape, or fabric tree wrap and put the sticky material on that wrap rather than straight onto bark, which it can injure. Then check for branches touching the ground, a wall, or another plant, since ants use those to get around the band. Once the honeydew stops, the film weathers and flakes away, and there is no call for a fungicide against the mold itself.The mold colonizes honeydew rather than plant tissue, so control begins with the honeydew producer and not with the fungus, and ant tending is what sustains that producer by displacing its natural enemies. Excluding ants therefore acts on the mold's food supply upstream of the fungus. Collar the trunk with heavy paper, duct tape or fabric wrap and apply the sticky material to the collar rather than to bark, following UC IPM, and close the branch bridges to soil, wall or adjacent plants, since a band with a route around it changes nothing. Expect a lag: a colony established before the band went on comes down slowly, and the existing coating weathers off rather than being cured.