Go back to Quadratec and order the GenII headlights for the TJ. The only difference between the ones for the TJ (H4) connector is the adapter from H13 to H4 with the anti-flicker pulse width modulator (that square block in the middle of the adapter). You already have that.
In fact, I would do one thing even before you buy new headlights - swap them from side to side. If the one on the left side is out and the one on the right on, you know it's not the headlights. It may be the PWM has gone bad. They are available separately. You did not indicate if it is the low beam only out or both high and low. If it looks like it's the adapter, swap them from side to side and see if the issue moves. If it does not, and you have swapped the headlights first, it's not the headlights.
Quadratec is now offering a Pulse Width Modulation (PWM) Adapter Harness that eliminates headlight flicker in 2007- present Jeep® Wrangler JK models. This H13 to H4 adapter stores power during "on" time which allows the lamp to utilize constant on energy thus eliminating the flicker.
www.quadratec.com
While they are shown to be out of stock until Oct 12. If you kept your original headlights, simply pop them back in until you get the new parts. It's hard to beat a $50 fix for the headlights.
If you have to, and don't have the original headlights anymore, you can drop a pair of standard H4 Phillips/Sylvania or other headlights in on a temporary basis with a set of KC-Hilites H4 to H13 adapters. The total would be about $40.
BTW the Quadratec GenIIs came with a 3 year warranty.
If my '17 Chief had not come with the Mopar factory LED headlights, it