Mine just started up with what sounds like it's probably the same noise. I describe it as a chirping that comes from the belts or water pump, only after it reaches normal engine temp and I only hear it when I pull into my garage or at drive thrus.
Pretty positive it's a bad water pump for me, as I read they are a known problem for the 5.7 specifically, yet it seems related to the 3.6 too.
Here's a video I found of a guy with a Wrangler 3.6 Pentastar engine that needs the water pump to be replaced:
It sounds almost identical to mine: (ignore the terrible stretched image and watermark. the noise is what i was capturing and you can hear it pretty clearly)
I reluctantly called my dealer and quickly asked a service tech about a weird noise coming from the belts and I thought it may be a bad water pump, to which the tech immediately tuned me out. They seem to do that when you come in saying you "think you know" what the problem may be. He was an asshat so I decided I'll live with it a bit longer.