Turn off printer.
Reboot computer.
Boot computer, turn on printer.
Take a screenshot of your print settings box.
I think if the HP is there, in your print settings box (in main menu), and you have no printing, then you need a driver.
Either the all-in-one hplips thing or custom or it exists somewhere.
Looking for drivers is step 2. Confirm that solus sees the hp in print settings.