Normally, except that this may not be a norm sit, several gentle cleaning cycles should do it-avoid any heavy duty cleaning runs for now. Remember, we don't know yet where the leak is--my worry is possible damage to the cyan inlet ink port to the printer. I'm running a 3880, a smaller brother to your 9880, the function is the same but the physical hardware is 'different'.
I assume your 'fuel system' is pressurized as is the 3880, therefore the ink is 'pushed' down the lines and then 'sucked' into the printhead. This is probably occurring but after some active printing, the ink is being blocked 'somewhere', and I now realize that there can be some debris in the cart exit port, the cart pressure inlet port or ink inlet port on the printhead.
I re-note that you were successful in printing several small test prints--now that sounds reasonable. In between test prints while you're sweating it out the 'possible' debris settles down--wherever it is. But when you retry the big pic, the debris is disturbed and starts building up blockage of
-this blocked location then drastically reduces the cyan flow or stops it. When you stop printing the debris 'settles' down. Nice!!
Could the debris be in the exit port of the cart--and then migrated into the input port of the printhead resulting in 2 active debris sites? Could the debris be located further in the printhead? The key note here is your printer mechanically and electrically is functioning-good! I think the problem now is to find the 'shmutz'. If this were a car brake system, you can easily unbolt and look-but it isn't.
The search path becomes clear-but how to for 'us' is not. A first step is to replace the cyan cart, if still no change, its possible that cyan cart had debris and spread its cancer-put it aside. It is quite possible that the check valve in the cart's exit port is flakey-let's hope. Now if the input port to the printer has debris in it--how do we clean it? Thirdly, if there is debris in the printhead-how do we clean it?? BTW, do not shake the carts vigorously, you will cause foaming.
What we do know is that the 'clog' is usage dependent--the longer the continuous print time, the greater the chance of a clog--in your case 100% in x minutes. I'm quite certain if you tried to print 10+ 4X6 at once. you wont make it.
If you are in warranty or have a repair contract, this is a job for Epson techs, now that you can inform them of the actual most probable cause.
BTW, what do YOU mean by posterization, if you are losing cyan during printing, your image should show a very warm cast but still have normal detail. Is this so? If not, what does your 'bad' part of the image look like, are you seeing blocks of pixels?
Let us know what happens when you use a new cyan cart.