The only "volatile" information in an unpowered modern CPU is any MicroCode software that is loaded in the CPU as an update to the hardcoded instruction-set of the CPU.
(This MicroCode usually contains security fixes and error-corrections for problems that were discovered after the hardcoded instruction-set was finalized. Changing the hardcoded instruction-set requires a very costly re-design of the CPU, which usually isn't done.)
This MicroCode is preloaded in the factory or, when the CPU is actually in use, can also be loaded/updated by the BIOS/UEFI of the motherboard or the operating system.
Regardless of how the MicroCode (if any) was put in the CPU, it is stored in a special on-chip memory that will hold its content, without power, for at least several years, possibly a decade or more.
If this stored MicroCode would somehow get corrupted the CPU will detect this on startup (as part of its boot-time internal diagnostics) and will simply not use it, falling back purely on the hardcoded instructions set. And then the BIOS/UEFI or OS can re-install that MicroCode after the PC is booted up.
So, if effect, your concern, while valid, doesn't really cause any problem in practice.