I had the same issues, i traced all the leads, i did an ohms reading on the scensor, bank 1 sensor 1 is the front manifold first scensor.
i replaced the sensor, 4 time 2 after market 2 OEM same thing kept getting P0135, changed the relay, checked the fuses, again checked all the wires, So i threw in the towel and took it to the dealer, they changed the sensor again, and again got the same code, to they hooked up the computer to my system. and found my computer is throwing a false reading/test. they are replacing the Computer on warranty.
Built not bought!!!
2005 Rubicon Unlimited, 5.5 inch custom long arm, 1 inch daystar body lift, River Raider Snorkel, Custom bumper/winch mount, sliders, tire carrier. 37x13.50 R17 Nitto Mudgraplers, Bushwacker flat fender. Custom 1/2 cab. Hyline hood vents. Custom tonnue cover and tool box. HID-X off road lights, LED driving lights custom rear bumper/m6000 rear monted winch.