All IL politics aside....
Lisa Madigan cannot ask for a continuance on the stay, unless she files for cert to the Supreme Court. Should she file for cert, she can then ask for a Stay of Mandate from the CA7. If they (the CA7) don't grant this (the reality is that they will), Ms. Madigan will then ask the Supreme Court for a Stay of Mandate (which will be granted). Those are the rules.
If the Statutes at issue (“Unlawful Use of Weapons” (720 ILCS 5/24-1) and “Aggravated
Unlawful Use of Weapons” (720 ILCS 5/24-1.6)) are not legislatively overturned, or made void/changed by other, intervening legislation, the Mandate goes into effect on June 9th.
Whatever else might happen on or after this date, as in local rules, no one in IL can be successfully prosecuted under those State statutes.
The above assumes that no law is in place by June 9th.