Does the phrase that one "keeps their head down" have a negative or derogatory connotation in regards to the person whose head is kept down?
I believe the phrase is derived from war, in particular, trench warfare where keeping their heads down meant that they were less likely to be shot.
The term can be neutral:
In the battle of rebuke between the cranky moderators, the average users kept their heads down.
It can be critical, suggesting cowardice:
When the CEO scanned the room seeking a solution, even the senior managers kept their heads down.
It might reflect smarts:
As the public panicked at the escape of the animals, the game wardens kept their heads down and waited for an opportunity to recapture.
It is contextual, but I think it tends to be used negatively, less so neutrally, rarely positively.