You overlook the fact that he passed a lie detector test. So even if he isn't innocent he certainly believes that he is.
Not all people who can beat a lie detector have conviced themselves they are telling the truth. If you are a calm confident liar you can beat it regardless of the skill of the person operating the equipment.
It detects that you are nervous so there are people telling the truth who still fail and people who lie who are not detected. A pathological liar can beat even the best skilled operator.
With that said a lie detector depends a great deal on its operator and the one who performed his has a history of mistakes. On TV they show tremendous changes when you lie but there are actually pretty subtle changes and what 1 operator views as deceptive another might not.
This is why a lie detector is used mainly for investigation. Often out of desperation to narrow down a large field of subjects. But on occasion that backfires and guilty people are cleared, an investigation drags on and years later they find out they had cleared the suspect already.
The only way Jeremy would ever confess were if there were something very favorable he stood to gain from it. The only benefit would be going from 0% chance of parole to a .01% percent chance of parole and being broke if he does get paroled so having to work for a living. I don't think anyone in his place would confess.