06 Jeep Wrangler X Bad Miss fire

First thing I'd do in that case is a compression check on all cylinders.
Then I'd prove that the injectors for 3 & 4 are actually not working when it starts missing.
Check the distributor for carbon tracking.
Monitor the spark to make sure it's not ignition related. Prove that it's firing 3 & 4 when it starts missing.
If that didn't solve anything I'd start ohming out wires between the PCU, injectors, and the O2 sensors.
I'd also become very friendly with someone with a similar year TJ and sweet talk them into trying the PCU in their rig.
 
First thing I'd do in that case is a compression check on all cylinders.
Then I'd prove that the injectors for 3 & 4 are actually not working when it starts missing.
Check the distributor for carbon tracking.
Monitor the spark to make sure it's not ignition related. Prove that it's firing 3 & 4 when it starts missing.
If that didn't solve anything I'd start ohming out wires between the PCU, injectors, and the O2 sensors.
I'd also become very friendly with someone with a similar year TJ and sweet talk them into trying the PCU in their rig.
I didn't think the TJs had distributors I thought they all went to coil packs
 
Mine most definitely has a distributor. If yours doesn't, and uses coil packs instead, I'd look there. Coils can work perfectly when cold and their internal windings can open as they warm up reducing the amount of voltage to the secondary windings, causing a miss or a total loss of spark.
 
I'll look at the service manual for the 06 tomorrow and see if I can find any commonalities between #3 & #4.
If you still have the original O2 sensors you might try putting them back in.
It's possible your PCU may be going into limp mode.
 
So the coil pack has three sets of coils, each providing spark to two cylinders. 1&5,2&6, and guess what - 3&4.
Change the coil pack before you try anything else, it's the only thing those two cylinders have in common that aren't shared by the other 4.
 
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The chances of all three of the modules that you've tried so far all having the exact same problem are somewhere in the range of slim and none. You really need to spend your time checking the existing components out rather than waiting on the big brown truck.
 
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I will say that if it is just cylinders 3-4 that clearly must be connected. My first thought (as already pointed out) is of course the coil pack.
 
You used the test light to make sure that injectors 3 and 4 are firing?