In a public place, there's no way that your spouse would be out of control. She would need to realize that she's under microscope with other eyes around so she may adjust her behavior accordingly.
In regard to your thought that maybe rumors are just rumors, I am going to say that 99.9% of WAS here have OM/OW in the picture. It is highly likely that W is indeed having an A with someone. This brings me to the next point. Would asking for the truth and making her swear on our children be out of line?
Don't act silly. Just calmly show her the book and ask if she had something to say about it. Quietly listen. Then say, "W, you know, we cannot be having any dinners or any joint activities going forward as I am not willing to have a third party in the M. You need to figure this out yourself."
With her saying she doesn't know if she trusts me and figuring she will eventually forgive me,
Simply say, "I am sorry that you feel this way. I am here if you wish to talk." The forgiveness part is up to her.
Validate, validate...it is how W is feeling at the moment. Don't try to discount them or argue back with her.
