I've run everything from 0w20 to 10w30 to 0w40 and it made no difference. The consumption varies between zero per 1k miles to almost 1 quart per 1k. I have a lifetime powertrain warranty and was told that when it hits 1 quart per 1k, they'll replace the engine. Not rebuild it, but replace it. It has gotten close to that level but not quite there. So over many, many miles and OCI's, I've found that it really doesn't care what oil is in it, it will consume it, or not, depending on its mood.
So with 158k miles on it, 118k of those have seen this. But it runs like brand new. So I just monitor it and add oil when necessary. Right now I'm about 3k into a 6k OCI with Mobil 1 EP 0w20 and it is barely using any at all. I'd love to tell you its that oil but it isn't. I just run that oil because the PDS shows lower phosphorus than most other oils, to theoretically protect the cats when it burns it.
I've had a mixed bag with the PCV, too. I once changed it and the consumption dramatically lessened only to come right back a few thousand miles later. I've probably changed it 4 times and that only happened once. The other times it made no difference. And all the PCV's moved freely when you shook them, so I don't think any of them went bad.
I've also bought a catch can but haven't installed it yet. My worry is that if it is going though the PCV (I'm not convinced of that) it will fill up VERY quickly. I may yet try that out of curiosity. I just need to fabricate a bracket.
Finally, a friend in the know told me that it was the pistons and block warp over time and you get piston blow-by. It isn't repairable. There are other theories on the rings and where the gaps line up and I've also heard that stuck rings can cause it. Again, the oddball thing is that it varies. I'd think with any of those issues other than the rings moving around, it would be somewhat consistent. Mine has not been. It's all over the place.
So what can you do? Just run the oil you like and check it frequently, adding when necessary. Oh, and you can run up to a quart over the fill line on the dipstick without causing a problem (in any FCA engine). I've been known to go 0.5 quarts over during periods of higher consumption.
Now the good news. If you monitor it and keep it topped off, these dang things run forever. I know of several with 300k plus and have seen many with over 200k. Keep an eye on mine, too, as the odometer keeps on climbing. Again, at 158k it runs like new. Sometimes I'd swear better than new. It hasn't lost any power over the years, either. One important thing I do is to change the spark plugs about every 40-50k. The gaps widen easily and I'll get a bit of pinging. 100k is just too long on these plugs. And I've run several brand of plugs. The ONLY plugs it likes are the OEM Champions. All others produced lots of annoying pinging. Even top of the line Bosch or NGK Irridium plugs. My engine rejected them within a few thousand miles.
Dan Grek did a trip circumnavigating Africa a few years ago. He built a Wrangler Rubicon up as a camper and put a Mercedes diesel in it. When testing it, the diesel blew. Being short on $$, he found another Rubicon with a 3.8. It never missed a beat and took him the 80k reliably, I'm sure on some loads of fuel that wasn't exactly the best quality. (I'm sure he filtered the heck out of it, but still....) It's a rock solid engine that goes the distance with proper care.
On the flip side, I have a friend that never checked his oil and his 3.8 consumed itself dry and siezed up. So even if consumption appears to be minimal, I'd check that dip stick every 500 to 1000 miles and top it off when necessary.
I've done the 6k oil changes as indicated by the OLM, to keep within warranty requirements. If I didn't have that warranty, I'd go a good bit longer, since it gets so much fresh oil added anyway. I'd bet if I went 30k on a change during a time of max consumption, an oil analysis would show it to be in excellent shape. Lol!