Spent quite some time struggling with the same. I came to the conclusion that there is an issue with the stock BIOS for HP's AMD R9 390X causing the driver install to fail and the system to hang, getting stuck in a reboot loop.
I have created a custom BIOS for the AMD R9 390X HP OEM. I used an XFX BIOS (XFX.R9390X.8192.150612.rom) and updated fan and other settings to be equal to the original HP one.
I have flashed it and the GPU is working fine now. AMD drivers were updated to the latest version "win10-radeon-software-adrenalin-2020-edition-20.7.1-july9", 27.20.2001.8002.
A link to the modified ROM below. Use at own risk.
HP AMD Radeon R9 390X 113-C67132-104 - fixed
You should be able to use the flashing tool on TechPowerUp to flash the ROM that I have created:
ATI ATIFlash / AMD VBFlash (2.93) Download | TechPowerUp
Let me know if this has worked for you.
Output from GPU-Z: