Oluf
Wild,wild speculation my friend. And for the record is the anti-ice system on the 777 not automatic..i.e.doesn't require "switching on" in the traditional sense.
Also i understand that at idle thrust adequate engine anti-ice is provided by the switch from low-stage to high stage bleed air. Otherwise no anti-ice would be available from high flight levels for long periods of time.
Just leave it to the AAIB who i'm sure will have considered this at some point QED.