You say "turns over and tries to fire" - does it backfire or catch for a second then stall or ??? Do you smell raw fuel out the tailpipe?
This should have been simple for a shop that has a diagnostic computer that can read a live data stream - call the shop and give them a heads up - have her run around and get the jeep good and hot and then go to the shop. Turn it off and then wait the appropriate time for the parts to heat soak and cause the no-restart condition - then the shop can plug in their diagnostic computer and watch the Crank and Cam positions, the Coolant Temp, TPS, MAP, and injector pulses would be a good start. If everything looks good then test the fuel pressure (doubt that's it since it restarts without fail when cold). Also it could be the ignition coil but usually that would happen in hotter summer weather when they get hot they can develop an open circuit in the windings - but there would usually be occasional rough running or a random stall too.
08 Rubicon Unl 4",35" KM2, F&R bumper, etc.
99 TJ - 4.0/AW4/NP231 swap 2.5/1.25 lift, 31" MT/R