I took my engagement photos in the conservatory 12 years ago and there was no permit fee. We did end up leaving a donation for the conservatory anyway, since it's something we do whenever we visit. Our donation was a bit more that time around though. I feel that if you aren't going to inhibit natural flow through the conservatory or have a bunch of equipment, the donation should be fine.
You can find photo permit info here: https://garfieldconservatory.org/visit/photography-at-the-conservatory/
Sorry no idea, call the park district
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