Incorrect, and it may depend on the company. A few years ago I washed out of a regional sim program during the 300hr new-hire days. I've since been successful at multiple ATP and Type rides for both 121 and 135 carriers. When hired at my next company I requested a copy of my pria, which is your right (FAA Form 8060-11A) and highly recommended.

It listed every required training lesson and the completion status. At the end instead of a line stating resigned during training (which is what the company told me at the time it would indicate), it indicated "Training terminated - performance related."

I was honest with future employers and they already knew the story, but be very wary of what a company that is terminating you is telling you. Often times it is not in your best interest.

Additionally a bad PRIA isn't a black-stain forever. It is a federal law that states that future employers are only allowed to request records for the previous 5 years.

The FAA website has a lot of info about the PRIA program, it is important that we as pilots understand how the system works so that it can not be used against us.
Pilot Records Improvement Act of 1996 (PRIA) Privacy Act Verification Letters (PA) Freedom Of Information Act (FOIA)