Maybe the initial question was specifically related to the 737 type of aircraft, ( vs. an axial thrust Eurofighter), Where the increased drag of a windmilling engine, on the end of a pylon, already causes a nose down pitch change.
So... ''True enough but to maintain the speed constant after reducing the throttle requires a pitch down, though not necessarily a trim change.'' Might be aircraft type specific only. It would all depend upon the position of the engine thrust line.