In order to confirm whether this is a bug I will need to perform some tests on my end. Would it be possible to tell me what is your current version of Sitefinity, so I can use the same version on my end? As far as I can understand the steps needed for reproducing this problem are the following:
1. Create a page with search widget;
2. Create search index on both environments;
3. Try to sync and check the search widget settings on the Prod environment;
If there are any additional steps that I have to follow, please feel free to add them.
Do you want to have your say in the Sitefinity development roadmap? Do you want to know when a feature you requested is added or when a bug fixed? Explore the Telerik Sitefinity CMS Ideas&Feedback Portal
and vote to affect the priority of the items