odd failure nd it could be related to a bad motor (since it seems to worsen when it is cold) as well as bad switch. I attached the rear wiper section from the SM and it will tell you how to check the functions for the switch etc. for your Jeep. Of course you also need to check the wiring. Jeep GC Rear Wiper System
“The rear wiper module electronic circuitry controls the switching of battery current to the rear wiper motor brush, which controls wiper motor operation.
The intermittent wipe and wipe-after-wash features of the rear wiper and washer system are both provided by the rear wiper module electronic circuitry.
The rear wiper module electronic circuitry also monitors the liftgate flip-up glass ajar switch and will park the rear wiper blade off of the glass any time it senses that the liftgate flip-up glass is ajar,
This is an interesting part and needs to be checked “However, if the ignition switch is turned to the Off position while the rear wiper is operating, the right multi-function switch control sleeve must be cycled to the Off position and back to the On or Delay position after the ignition switch is turned back On before the rear wiper will operate again.”