How would a sociopath empathize at all? That's the primary skill they lack.
So a falsifiable hypothesis might be:
Sociopaths have no empathy at all. Their perception that some other empathizes with them or that their empathy with the other is recognized by the other is some random function based on their internal dialogue and whether they think they get a better cookie by stating that empathy exists when prompted by the experimenter. By comparison, people who score higher on some Empathy Scale (tm) state that empathy exists at a higher or lower rate than the sociopathic control group.
HTH. I don't claim that hypothesis personally, as stated that's what I was maybe hoping to find. I'm just a programmer, whether that's programming computers or minds is sometimes unclear.