I was thinking about the 7 days theme as well, but I figured that long resting doesn’t actually interact with any time deadline. After all, if you thought you were able to complete the objective after taking your “last long rest,” but made a mistake, you could actually put your game in an unwinnable state. No way to recover HP or spell slots, and no way to win the next sequence of required fights. Unless you had an earlier save, you’d have to start the entire game over. I don’t think they would implement something like this.
Unless of course seven days did pass and there was as an unavoidable story choice made for you forcing you into a particular path. I.e. the demon who said “when you’ve exhausted all of your options” he will solve your problem. Now you’re indebted to a demon for taking too long.