I have a HP OfficeJet 8720 printer which does not recognize non HP manufactured cartridges anymore. Those from Amazon, etc.
It gives the message, "Supply system Problem". Try replacing the affected cartidges with a differntnon-empty cartridge.
Does anyone know how to go around that, or do I have to just buy or lease another Printer?

Never update hp printer firmware, they block the chips constantly of the knockoff ink. Disable them if you have them turned on.

You are sol for the ink you bought. Gotta play the whack a mole game to see if the ink you get has an unbanned chip before you keep it from Amazon.

I trashed my brother printer for the same reason years ago and went epson ecotank. I recommend any Inktank in general now over cartridge based.