PS4 stuck in some kind of boot loop?

My friend's PS4 boots, starts checking the disk, and then asks for an update to 4.72 or later.

Using an usb stick, it'll just say it can't read the file.
Going into safe mode, and then selecting update from an usb stick, it'll read it all, say it'll reboot and then starts up trying to read the disk again.
Safe mode, download the update over the internet does the exact same thing.
Rebuilding the database also doesn't work, it'll just go back to checking the disk after rebooting.

Is there a way to fix this without completely reinitializing the PS4 (and losing the saves) or are we out of luck?