I honestly did not know anything about the Illinois Family Relief plan– which includes the current 10–day period of lowered tax on back-to-school supplies. My research led me to THIS ARTICLE which not only has a full explanation but offers resources to contact them for details.
From what I can see, this only applies to 2021 tax returns. I do not know if it will be extended to 2022 returns. If it’s a one-year program, and your ex gets to claim your daughter for all those benefits included in the plan, it would be nice if he shared some of the windfall with you!!