they can't actually demand proof in almost every case. The only legal grounds are a non-compete clause in your current contract plus SVP level or above. Publicis just wants to verify you're not hopping to a competing agency or client, like an agency on the same account or a rival brand in the same category. if they decide you are, they can try to block the move.
Should that threat ever show up, hit them with something like "turning down this opportunity would cause financial hardship, and if you have further questions, you're welcome to speak with my lawyer." Whether that lawyer exists is between you and your conscience. they back off real fast once legal gets mentioned.
Outside of that one scenario, you owe them zero details about your new offer or employer. It's at-will employment, and it cuts both ways. They can terminate you tomorrow with no explanation, and you can walk out the door the exact same way