What happens if the input voltage of a low dropout regulator is under the rated minimum voltage?
I found this LDO regulator that pulls more current to maintain the output voltage until the input drops below a certain value, then it shuts OFF. This is perfect for battery operated devices who's battery voltage drops while in use, but at some point the battery is too low, so it should shut off. Part number: LT3645.

Installing a rectifier in the output makes it an 'open' circuit 'OR' gate, neither sourcing or sinking current if the regulator is OFF.