Mark the edges of the damage and see if it progresses. If it does, cut it back to healthy tissue. If it doesn't, leave it. Those aborted tips fall off on their own eventually. Either way you'll have some new branches emerge from the leaf scars on that branch

agree. my preference is to let the plant naturally wall off the area. However you must be observant for signs the plant has not been able to wall it off. If you cannot invest the time then go ahead and cut off the area leaving some leaf scars as that is where the plant may branch out. If you want to remove the branch all together then cut back near the collar of the branch.

Where and how have you been storing it?

It's been in an unheated, unused bathroom all winter. Just let it dry out in the pot.

I'm fearing the worst for this one, I see black tip just starting on the other two tips, it hasn't woken up except for a tiny bit of sap on the other tips. Fortunately, it came from OSH, I can get it replaced when they get some in :)