It depends on who exactly they kill, or what they empathize with. They might start out as a very sensitive character who can’t bear to pick up a flower, so it doesn’t even have to be a murder. The more they do it the more fucked up they become but if the fifth murder is a loved one, it’s more morally reprehensible, or depending on their moral compass, a child, an old person, a woman, a teacher, a cop, all this can skew the balance either way. So, in general, yes, but not always