Hi Neil, glad to see you here!
Many refill carts, including the R3000, draw ink from the bottom of the cartridge. The longer pigment sits still, the more pigment particles will settle to the bottom (this is true with any pigment ink, including Epson). My recommendation of weekly cartridge agitation is based on how I maintain in-suspension ink with our pro model production printers, which I can easily shake to agitate carts without removing. My weekly schedule is probably overkill, but keeps me from wondering when it was last done because I'm working with a LOT of printers every day. My routine is to have a piece of masking tape on each printer with the date each printer was last used, so I can keep track of the use and make sure they are all exercised once a week to maintain in-suspension ink in the lines and dampers, and always shake the printer/carts a few minutes before turning a printer on. I also have a plastic sleeve attached to the side of each printer, where I keep data specific to that machine in regards to service, ink, carts, and other notes I want to track. If a printer goes longer than two weeks between use (I am very busy and as I said, am working with many printers), I shake the carts, then do a few cleaning cycles and print a 21 step strip to measure and evaluate the output linearization for accuracy before printing images.
For the R3000 and other desktop model printers, I think bi-weekly cartridge agitation is sufficient, though weekly use of the printer is still our recommendation to help prevent ink settling in the lines and drying of the print head. With the R3000 and 3800/3880 printers that have internal ink lines, I think it's best to remove and agitate the carts with the printer OFF to help prevent air from entering the ink lines. As long as the exit chambers are correctly filled with ink and carts are not kept out of the printer for a long time, I feel this is safe. It may be possible to have the R3000 or 3800/3880 on a cart with wheels to shake the whole printer as we do with the pro models on wheeled stands, but I have not done this myself so can only assume the left cartridge side of the printers would only need to be "agitated", and it would have to be several short/quick movements to really shake the ink in the carts. You would obviously want to be very careful if there's an external waste ink bottle attached to the printer, to avoid spilling. My personal experience with our R3000 printer is that I print often enough that I have to refill the carts about every other week, so just shake the carts at that time, and have not had a problem with ink settling or air in the lines.
I hope this helps, please let me know if you have further questions or there's anything else I can help you with.
All the best~ Dana