Aircraft Mechanic reported a problem with engine thrust reverser not stowing correctly and departing the aircraft during an engine test.

Narrative

At XA:30; one hour before my shift was to end; my crew chief asked me if I could help run the engine on Aircraft X. The 3 mechanics that were on that aircraft had a transit check and a borescope. I was to run the aircraft after the borescope was completed to check for leaks on the drive pad. We conducted the leak check and no leaks were noted. We shut the engine off and I performed the operational check on the thrust reverser through the Multi-Purpose Control Display Unit. The test allowed me to deploy the Thrust Reverser but when I went to stow the Thrust Reverser it stowed almost fully closed and the test failed.I tried once more to repeat the process with the same result happening. The Thrust Reverser would extend but would not stow fully closed leaving roughly a 3 inch gap. The test failed again. We repeated the test a third time with the same result. It was at this time we decided to start the engine to give the force to fully close the Thrust Reverser. When we started the engine the engine created a large noise replicating the sound it makes when it goes to takeoff at which time I shut the engine off but not before the outboard fan cowl dislodged from the engine in a forward trajectory. I believe the above event caused an intense reverse airflow causing the outboard fan cowl to be dislodged from the engine. I take responsibility for the accident that happened; but other contributing factors happened as well. I was getting pressure from my crew chief who says he was getting pressure from the manager to complete the job before our shift ended. The event happened 13 minutes before the end of shift. I also felt there wasn't enough manpower provided; and I should and wish I had said and done something about it. I learned an immense amount of what I should have done and what I will do in the future.