As Arpaio counsel Jack Wilenchik tells Arizona's Law/Arizona's Politics, the appeal - currently in the 9th Circuit - will continue despite the adverse decision from the Supreme Court.
"The Supreme Court just doesn’t have enough time to correct all of the Ninth Circuit’s mistakes. Appeal goes forward as normal except that the USA now has two sets of prosecutors who don’t agree with each other because the Ninth Circuit ordered that it be so."The Ninth Circuit appeal was on hold until the U.S. Solicitor General informed the court in January that the government would not appeal the decision. The briefing schedule continued despite the former Sheriff's decision to ask the Supreme Court to get involved. Last week, both the Special Prosecutor and the Department of Justice filed their answering briefs with the Ninth Circuit, and Arpaio's Reply is due later in May.
*Arizona's Politics and Arizona's Law are sister publications. In situations such as this, there is overlap.