Have been through a few Mopar and non Mopar versions looking for results.
I had the OPDA timed with the old timing chain to TDC.
Once the new timing chain was installed, took it for a short drive without any further changes. It started throwing 0016 codes (in addition to the 0344 and 0340 codes).
I then retimed the OPDA to TDC and all codes disappeared and haven't returned. No check engine light has shown since changing the timing chain and retiming.
A sideline is that the engine is now idling smoother, it's also quieter at idle and revs like a newly rebuilt engine.
FWIW when the problems were happening I talked to Wrangler fix and he said it probably wasn't an ECU issue.
It must be a timing issue. 0344 now popped up. Do you have good instructions on how to retime to TDC?