If you've had the JK serviced at the dealer all of your service history is stored on their computer system. The only time you really need the other proof of service is if you've had it serviced elsewhere. There is a leeway on the service interval, that if you approach them in a realistic manner and not go in guns blazing as some people are wont to do, then you are more likely to get a good result.
The extended warranty is actually a bit of a rort as it is not warranty at all but an insurance against failure, so the decision to reject is more like the insurance company trying to keep your, sorry their (you've already given it to them) money.
So you'll find the dealers far more likely to carry out actual warranty work with a bigger tolerance as the money to fix comes from the car company and not from the insurer who has a different agenda.
Good luck if it is stuffed.
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