Hello and welcome to Just Answer!
Lets start by checking the fault code that is setting!
Try this - Sit in the vehicle with all doors closed and starting with the key in the off position turn it to ON, OFF, ON, OFF, ON within 5 seconds and leave the key on and watch the digital odometer to see if it displays a fault code. It will be a P followed by 4 numbers example P0700. When it is done displaying codes or if there are no codes it will say done.
I have never seen a timing belt/incorrect timing cause a P0344 fault code and I wouldn't be concerned about the belt at that mileage!
This code can be set by an intermittent glitch from the crankshaft position sensor!
I wouldn't suspect the crank position sensor just yet! The PCM compares the crank and cam signals and if there is a glitch in either sensor it can cause the P0344 code and not cause any other codes especially if it is intermittent. I would have a look at the target magnet behind the cam sensor to see if it has come loose and moves around.
This type of problem is very difficult to diagnose without an oscilloscope! An oscilloscope is able to display the signal on a graph and you can check for missing/erratic signals from both the cam and crank sensors. You could waste alot of time and money by using trial and error. You can also perform a wiggle test when checked with an oscilloscope to see if there is a wiring problem.
I would suggest taking it to someone that has an oscilloscope and have them check the signal, if you want to diagnose this properly!
Let me know!
This is not a common problem, but I have seen a crank sensor cause this! I have also seen a loose or cracked target magnet cause this! I can't recall finding too many wiring issues with these, they have been good that way! Fault code P0340 is far more common than P0344!
You are very welcome!
A dual channel lab scope works the best, XXXXX XXXXX you can see both cam and crank patterns on one screen and can pick up glitches easier!