You have the exact same problem I had. The root cause is the flipper glass seal. There is a TSB for this but they have to run your vin to see if it applies to your vehicle. Also, I'm not sure if it will be covered due to age. You'll need the gate latch and/or flipper glass latch as well as a window seal and an additional piece of seal material that looks like a rain gutter that goes across the top of the glass and diverts water to either side. This seems to have been a common problem up to 2010. Hopefully mopar covers your parts.
2010 Liberty KK Sport, deep water blue, weathertech in channel vent visors, weathertech digital floor mats, Detoures backbone winch mount w/smittybilt XRC 10 comp winch, rigid industries dually led lights on smittybilt fairhead mount, Rocklizard skink supersliders and Gecko safari rack, cobra cb w/48" firestick antenna, general grabber at2 245/75-16, OME heavy duty springs, OME shocks, about 3" - 3 1/2" of lift, Detoures Knucklebone spare tire and Jerry can mount, mopar skids, class 3 hitch w/7 pin wire harness, headlight covers, 4.10's and Detroit TruTracs F&R