If you look at an HP cartridge, the connector side, the side that plugs on to the ink channel in the cart slot, you will see something similar to the head connector at the other end of the ink channel where the head is connected on in the heads carriage. Between the two connectors at either end of the ink channel there probably is nothing else than a long tube. I doubt there is a valve in between. The basic ink supply is done by the membrane pumps in the carts themselves, actuators in the printer's cart slots push against the soft black rubber ball on the carts and a small disc valve inside the cart makes that a pump delivering ink when demanded to the head ink buffer.
You have used a bulk ink system with I guess third party inks. If that system does not have a pump system but relies on gravity only it could be one of the reasons you have that issue now. Probably an ink channel that does not have flow anymore. Third party inks that have a pigment settling faster than the OEM inks is another factor.
As written before try to pull ink from a cart through that ink line by using a syringe that pulls a vacuum at the head side connector. If the channel is blocked or has too much air in it you can solve it that way.
You have to do it with this information. Someone with some DIY skills will be able to visualise how it works and deal with it. Otherwise it may not be a good idea to do it yourself.
Met vriendelijke groet, Ernst
Dinkla Grafische Techniek