What a conversation barrel? What's that? Dose it talk to you? LMAO. Just kidding!!! Yes I would trust my life to a Conversion barrel. If that's all I had. I've had a few LWD barrels 40-9 and they have always worked for me.
I would not, unless the gun had been fully changed to the caliber, ie magazines, extractor and ejector. Any one could fail at any time and that's never a good thing to trust your life to. It does matter if you've got a 1k failure free rounds, it only takes one. The only exception would be if it were converted to a caliber in the same family. .40>.357Sig, for instance.