I'm not so sure -- J. Kavanaugh's stay remains in place pending appeal or until SCOTUS acts on Cert Petition if one is timely filed. NPN can take the 8th Cir. to SCOTUS -- it will likely request a Stay pending filing a Petition.
Also, for confusion sake, this is not going to be subject to the Supreme Court's stay. The Eighth Circuit remands with order to dissolve the TRO and enter a permanent injunction. This is like what the First Circuit does. It stayed its own opinion for 7 days. Then, Missouri First!