The NY appeals court must intervene to act immediately; otherwise the federal court will have to act - or else. Normally the state appeal process can take months; that is inadequate when a leading candidate for president is threatened with being sequestered and prohibited from campaigning.
I think the appellate process is little likely to "toss the verdicts"; that normally occurs with copious paper work and detailed legal cause for dismissal. I doubt the appellate courts will take on that burden with an emergency appeal. The emergency is obvious - the end of the American Constitutional electoral process. To my untrained mind, a 'stay' is most likely so that the Trump presidential campaign can proceed and later time can be spent on documenting unethical and/or unconstitutional legal abuses.
It is one of the most critical crises in modern history, perhaps ever in our history. Democrat attempts to circumvent the normal electoral process in order to end all opposition to their permanent power shows what criminal traitors they consider themselves - and they do not care that they are perceived that way.☠😠
The “appelate process” is a murky, slippery subject. I am hoping that NY’s highest court will make a ruling soon. If the NY courts mess around until September, that will put SCOTUS in an uncomfortable position. SCOTUS already denied Jackass Smith’s motion to expedite, and Roberts and other justices don’t want SCOTUS to appear inconsistent.