Remain down can be synonymous with several phrases, including stay low, keep grounded, and lie low. These phrases imply a sense of caution and a desire to stay hidden or avoid attention. Other synonyms for remain down might include "stay put," "stay out of sight," or "keep quiet." The concept of remaining down could also be associated with concepts like humility or patience, suggesting the need to stay steadfast and focused over a longer period, even when circumstances or other factors might be pushing one to act more quickly or assertively. Ultimately, the specific synonym used will depend on the context of the situation.