Have a similar problem, P0673 (2012 with 100K miles), replaced all 6 plugs, still P0673, replaced the control module (I did remove the coolant pipe) and the code went away for 3 weeks and then came back.
Now what? No code for 3 weeks means that the controller was most likely the culprit but how likely is it that a new controller (genuine BMW from getBMWparts.com) goes bad within 3 weeks?

It finally got warm enough to take my car back apart to check the wiring on the control unit and found a small amount of dirt in the connector, cleaned it out and now everything works.
I was never able to clear the faults. It is hard to believe that the control unit would fail that quickly.

FWIW I used the free OBD Car Doctor app to clear the faults after replacing 6 glow plugs and the control module. The faults would not clear when the "key" had been inserted or after the engine was running. To clear the codes, I had to insert the key and press the start button without pressing on the brake.

Funf Dreisig

great advice to replace module

https://youtu.be/dXGnK8UVTF0