We'll see. This was a federal mandate before it was overturned
It was a federal mandate created by Biden, which is exactly the reason why it got overturned. Private businesses and institutions would have a better chance of the mandate sticking if it comes from them instead of Biden. Although, in the case of private institutions, we're seeing instances in state governors are interfering to prevent the mandate from happening at schools.
That's where the real fight and lawsuits are going to take place.
