The printer is designed to print when the ink is low, but once the ink is completely depleted, the printer will also stop printing. The reason behind this is because the depleted cartridge will damage the printhead of the printer if tried to print.
Also, there is no way of tricking the printer about cartridge being depleted. As far as I understand, the ink level is either measured by using a microchip in the cartridge.
Step 1: Check from the printer software on a Windows
Check estimated ink or toner levels for a printer connected to a Windows computer.
Turn on the printer and make sure that it is in a ready state and is connected to the computer.
Search Windows for HP, and then click your printer name from the list of results. If HP Solution Center displays in the results, click it.
Estimated ink or toner levels display on the home window of the printer software.
Figure : Example of Estimated Ink Levels in HP Printer Assistant
Figure : Example of Estimated Ink Levels in HP Solution Center
Figure : Example of Supplies Status in HP Device Toolbox
If you could not access your ink levels, continue to the next step.
For more assistance, please follow the steps suggested in the support document for - HP Printers - How to Check Ink or Toner Levels
HP Printers - Cannot Print When One of the HP Ink Cartridges is Empty
Hence, please use all new cartridges and check the printer functionality.
You can employ the steps I provided in my above message to limit the use of color ink cartridges.
Keep me posted, as I (Or a trusted colleague) shall follow-up on this case to ensure the concern has been addressed,
And your device is up and running again, only in case you don't get back to us, first!
That said, If the information I've provided was helpful, give us some reinforcement by clicking the solution and kudos buttons,
That'll help us, and others see that we've got the answers!
Good Luck.