I believe this might be a big key to the problem. Some cats don't like sharing boxes or using dirty boxes. I think if your mom is serious about correcting the problem she should start by having at least one box for every cat. I'd even recommend one extra.
Because this has gone on for so long it may take retraining Jasper to the litterbox - by confining him to one room with box until he gets back into the habit.
I am dealing with an inappropriate pee'er in my house and I know how frustrating it can be. The others are right, there is medication that can help. As soon as I figure out who the culprit is in my house they will be going on Prozac.
If your Mom isn't willing to increase the number of boxes or to medicate Jasper I'd say he's probably better off in a new home as long as those people know what they are getting in advance. I really hope your mom tells them about his peeing so he doesn't end up being thrown out or in a bad situation.
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