I have the 20" wheels and want a replacement for the OEM Forteras. 9K miles so far on the car and tires, and there's noticeably visible wear on the fronts so I'm not happy about that. I need to rotate the tires, and I plan to do that soon.
I'll keep those tires until they wear out but then will want a 20" tire for mild offroad use: a more aggressive tread, stronger (more resistant to punctures / sharp rocks), and taller sidewall height so when I air down I'll have a smoother ride, better contact patch, and more height to protect the pretty OEM wheels (20 X 8, 5-spoke) from getting scratched. I don't care about a wide tire, in fact I'd rather go with a taller and narrower tire.
Ideally, a somewhat quiet tire so it's not obnoxious on our long highway road trips. I do not want to lift my JGC. I'll keep the suspension at the stock height. Oh, and a tire that gets good mileage too.
So yeah I want it all.

In this order my priorities are:
- taller and skinnier 20" tire, to fill the wheel well as much as possible on the stock suspension setup
- more aggressive off road tread (an A/T should do, I don't need mudders)
- quiet(ish), and gets decent mileage (50K would be great)
This will narrow my options significantly but I'm pretty sure I can find a tire to fit. I already know I'll go outside of what the local tire shop says will "fit" on the rig.