Cars with the 6 cylinder N52 engine are prone to having the fault code 2EF5 which seems very similar to the 2EF4 info you got. I talked to a BMW Service Manager & he told me that the N52 error relates to the Map Thermostat too & that it (the thermostat) allows a change in direction of the flow of cooling towards to cylinder head which in turn helps aid a reduction in emissions. This code means this function is no longer working. He said that it does no harm to drive with 2EF5 & he also told me that they have seen loads of them in the workshop with this code & they don't do anything with them. He says that 9 times out of 10 it is a thermostat fault & a replacement will fix it, the other 1 out of 10 is down to a wiring fault.
He told me this off the record.
I hope this helps.