Booting Linux on the PS4 has been a possibility for a while now, and many users of firmware 1.76 have been able to try it. This week, developer valentinbreiz has managed to port the existing 1.76 payload to firmware 4.05, and has shared his compiled files.
Disclaimer: at the time of writing, I have not tested this yet. People report that this works, but booting Linux might be hit and miss due too ASLR.
Note that you can also compile everything from scratch yourself, instructions to do that can be found at the source below.
Copy the payload (ps4-Linux-Loader.bin) to the payload folder of ps4-exploit-host, then while making sure your USB drive is plugged into the PS4, send the payload to the PS4 like you usually do for other payloads. Linux should start.
Scene member @Darbnes has posted a video confirming this works fine