2EF5 (also 2EF4) are thermostat codes. Given you have overheating too I suggest replacing the thermostat asap.

If the water pump or tstat fail the fan defaults to fully activated. I would continue with
fan diagnosis. Can it be activated?

       check for 12 volt at pin 2 of the cooling fan connector red/blue wire . if missing there is a blown fuse f37 60amp located at the power distribution box right side of dash. if good check for ground pin 1 of the cooling fan motor connector brown wire if missing ground it anywhere you like but its located at the right front wheel well. the dme or engine computer grounds the black/green wire pin 4 of the cooling fan motor connector based on the readings from the engine coolant sensor but the thermostad needs to be checked also  as the coolant sensor may not be reading correctly. don't hold me to this in this particular vehicle but it has happen on other vehicles one more thing to keep in mind with electric cooling fan in any vehicle its   the radiator cap

I`d check themostat/s operation.
Most likely the main thermostat in the engine.
The dme monitors it, I often see the sensor part fall apart after 100k or so.
As ede65 says the dme is confused and turning the fan on.
If a car is overheating and it is not loosing coolant, change the themostat.

Edit; dme NOT turning the fan on