No, the software will not record if your phone is shut down completely. If only the screen is turned off, then it might be possible to run the recording while the software is in the background mode. Mobiles operating systems have a feature that allows the app to run in the background while doing other tasks. This is possible when the developer implements this feature. I know it's possible on Android, but for iOS it's unclear, and it's best to ask the developee.
I have checked, and it appears it's impossible to contact the developer, because the support link listed in the App Store is broken.