Tony, the chain being slack... there is a little spring there, and it just to keep the slack off when the engine is standing still. It is not until you get oil pressure that the slack is gone.
But, I see they sell those things new, so maybe they got so worn that either the oil is peeing out, or that they get stuck somehow.
The injectors does not cause the chain to get slack, that's a piece of crap. But, you knew that already... if someone think diesel injectors can slacken a timing chain then I guess I think they have had their head down the paint jar for too long.
What slackens it is just normal wear. I wonder what the opening pressure on the injectors are, cause if it is too little, it will start smoking.
I would getting that checked first. If the nozzles are not done right the engine can behave like a real lunatic.
That said, if the nozzle pressure is set way over what it should be the increased pressure tends to wear out the pump fast, and in that sense can cause more wear on the chain. But the pump will most likely go before the chain.