Apologies for my limited aerodynamics (remembered from the Air Cadets years ago) but surely flying nose down will result in a lower angle of attack (alpha) and therefore less drag.
It will of course result in less lift which could be recoved with more power.

More power meaning a higher fuel burn and less efficiency! 🙂 Airliners tend to cruise with the nose up 2 or 3 degrees as this is generally the most efficient angle of attack for them to fly. That’s as I understand it, anyway!

Paul