What do they have to show, then?
The Court of Arbitration for Sport is not a “real” court — it’s a binding arbitration board whose parties must mutually agree to abide by its decisions. Unless there was some violation of the law, why should the Swiss Federal Tribunal care if an American doesn’t like the ruling she previously agreed she’d abide by?
It doesn’t stand to reason that, actually, CAS doesn’t have the “final say” in these cases, the Swiss Federal Tribunal does.
Interestingly, Caster Semenya tried a similar Hail Mary appeal to the Swiss Federal Tribunal and lost. You can read their ruling here, but the Tribunal explicitly says in their dismissal that they didn’t have grounds to vacate the CAS ruling in her case.
“The SFC emphasized that its competences in the case were limited to examine whether the CAS Award violates fundamental and widely recognized principles of public order, which, as it clarified, includes the prohibition of discrimination, certain personality rights of athletes and the notion of human dignity”
If that description doesn’t apply to Caster Semenya’s case, it certainly doesn’t bode well for Shelby Houlihan.