I am clutching at straws here, but have you considered what might be termed "rocket effect".
In a rocket engine all the thrust comes from accelerating the mass of fuel, there is no airflow.
With a turbofan most of the thrust comes from accelarating air, but there must still be some small proportion of thrust which comes from accelerating the fuel mass.
Now engine anti ice makes the engine less efficient, therefore more fuel flow for the same gross thrust therefore more "rocket effect" therefore less airflow required to achieve the same gross thrust?
(disclaimer: this is pure speculation and may be complete nonsense)