The man advises the woman to _____ her head down regarding the arguments in the staff room.

The correct choice is "keep." In the context of this sentence, "keep your head down" is a common idiomatic expression that implies staying out of trouble or avoiding attention, particularly in a situation where there are conflicts or arguments. By advising the woman to keep her head down regarding the arguments in the staff room, the man is suggesting that she should refrain from getting involved in the disputes and maintain a low profile to avoid exacerbating the situation or drawing attention to herself. This phrase is often used in work settings or situations where conflict is present, highlighting the idea of self-preservation by not getting engaged in the ongoing disagreements.

The other options do not convey the same meaning as this idiom. While "hold," "put," and "let" can be used in various contexts, they do not carry the same connotation of staying uninvolved or avoiding further conflict.