I am really not certain what you have in the Cyan color position of your printer. But it is not a shade 2 ink according to your calibration mode. All the other inks are very warm and are sequentially in shade order. The Cyan position seems to be using a shade 5 Neutral ink. But I do not see any new ink purchases on your customer record other than a shade 1 black in Nov 2014. All the other inks appear to have been purchased in April 2013. But this ink in Cyan position which should be a very dark shade 2 looks instead like very light ConeColor LLK or a Neutral Shade 5. You have purchased ConeColor LLK but not Neutral Shade 5.
The presence of such a light ink where such a dark ink is expected does explain why your system is printing so light in the shadows. And why your linearization is so off in the shadows. Why your system is so dark in the upper regions is more difficult to trouble shoot. But - with one ink incorrect in the system there is a possibility that this system has been sitting idle for a long time at some point and all the inks have been subjected to changes of some sort from pigment falling out of suspension. That is my best guess. I think that you have two things going on at once. A mis-filled cartridge in the Cyan position and a printer that has sat with pigment ink in it for some time of inactivity.
Normally when a printer is behaving like this we instruct to remove the cartridges and shake them gently for 15 seconds. This will re-suspend the pigment. You can not shake the ink inside the printer's ink dampers and ink lines. You must push this stale ink out by performing three POWER CLEANS or an INITIAL INK CHARGE. However, I am suspect that you may need to replace the ink in the Cyan position as I do not think it is Warm Neutral Shade 2.