Does the phrase that one “keeps their head down” have a negative connotation

Question

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?

Answer 1

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.