I don't know if it's slam dunk case or not, but it won't turn on a public/private distinction. States have a lot of latitude in establishing the eligibility criteria for their public school athletes. Since most states have these determinations made by a governing body rather than a statute, private schools have long had greater latitude because they can always exit the state leagues in favor of their own if they find the rules onerous.
Judging only from a mass media article (a poor source of information for legal analysis), I would say the case is about whether the governing board overstepped the authority given to it by the state legislature and/or made a decision contrary to law.