Wayne, excellent advice!
Damper, maybe if there is a nasty pocket of air blocking that channel.
Check pump cap seals to printhead.
Initial fill for VLM is much like a Hail Mary, but usually resolves air pockets. And doesn't require opening the machine.
More advanced, not recommended if never serviced these machines:
Disconnect damper kit from print head and disconnect the tube junction at ink bays. Using a syringe, push warm water from the ink bay all the way through and out the damper kit; While redirecting the waste water to a bucket. Then remove dampers, inspect and flush/ replace if needed.
Remove printhead, inspect for damage, clean platen very gently by running warm water (pure bottled water) over the nozzle plate surface.
Remove printhead first, soak platen in ink drawn from one of the cartridges while flushing the ink delivery lines. reinstall printhead last after wiping off the ink using warm bottled water.
Reassemble, run initial fill, nozzle check, initial fill if needed, nozzle check.
This procedure, while very risky to non-service techs, does usually resolve. If unsuccessful, it's 95% due to a failed printhead and/or failed pump cap.
To further complete this procedure, remove ink bay(s) and flush with warm, pure water.
Note, while flushing the system; I've found the most resistance pushing through water due to a bad damper or a kink in that ink line. Sometimes the plastic mounting brackets for the ink tubes apply pressure on the lines, starving the system of ink. The ink tubes sometimes expand. Removing these brackets sometimes recovers the system.
But your problem is the system ran out of ink. Flushing the system has a good chance to resolve, pending the printhead is not permanently blocked. Which brings me back to checking the pump cap connection; and flushing the system of dry ink..
Calling Epson will just throw a bunch of parts at it, at your expense after warrantee.