The settings are saved when the command AT+COPS=15 is used.
Also the AT+CPWROFF does a save shutdown, it write some things in the module.
The next time you power up, the settings are checked and it will try to re-connect.
The library will take care of the reconnect, you are known in the network, so it should be faster.
If you use the correct settings then the library will check it, don’t reset it, and just re-connect.
If you use mno profile 1, this will be an issue.
mno profile 1, means: read from simcard and set the correct profile.
In the library it will check the mno value, this will never be 1.
For testing purposes this can be used, but for the final application you should set the profile to one in the mno profile list.
So in short, yes you must re-register but also some things are still known.