Only if it was a very high threshold - so high that you could see all the story/relationship beats before the penalties kick in. Otherwise you're making players choose between exploring major aspects of the game or maintaining their agency. Also it would disproportionately penalize good-aligned (or, at least, non-Absolute-aligned) characters.

The best way to solve the urgency problem, IMHO, is either to decouple story/relationship beats from long resting, or to make it more clear earlier on that you have more time than you think.