Hi ravekiss
It's a hard decision you have to make. It is finrot, which is a bacterial infection, but a very advanced case. Once their barbels are damaged that severely, its chance of survival is not too good. If it is cured, however, the barbels can grow back enough for him to get along.
The corys I've had like this needed intensive treatment to pull them through and it took some time. A mild case might clear up by itself simply by having an opportunity to live in clean and fresh water, but I don't think this one will. Look how skinny he is; that's not a good thing either because it means he has used up his reserve strength, so to speak.
Who will be taking care of your fish while you are gone?
If they will continue the medication, you might want to keep him and give him a chance. You can do the salt dips for a few days to get him off to a good start on the treatment. If he gets cured, he will be a nice fish for your aquarium.
BTW, I can't tell from the picture, but is your peppered cory full grown? I'm asking because there have been some wild caught ones around and if this is one of them (as opposed to being tank raised) you might want to get a few more. Their colors are much nicer and they are stronger fish than the usual ones available.