Green will probably be on the staff next year, he's under contract. The only way he wouldn't be back is if he and the team mutually parted ways like Nasreddine did at the end of 2020. I think they'll ask him if he'd like to come back as an assistant next year, and if he says no they'll release him from his contract.
That being said, if he was the "coach in waiting" and they hired him specifically to be the heir-apparent and slide in after Lindy was fired, they wouldn't have named him interim. Sometimes teams fire their coach and name a permanent replacement immediately, it's not that uncommon. Of the 7 coaches fired this season only 4 have been replaced by an interim. Roy, Hynes, and Knoblauch were named permanent HC with no strings or conditions.
I don't think making the playoffs, however unlikely, is enough for him to be promoted. They team has not played particularly well under him.