Can you, please try converting the Virtual Directory you have your site installed to an Application in IIS (please see the attached screenshot). The problem may occur because of the workflow that is activated when you try to publish something, and it's checking whether you have the proper rights to publish the item. However, since the authentication cookie is not the proper one Sitefinity is asking you to log in again. The proper way of handling this is to add an Application in IIS where you can install your SitefinityWebApp, not a Virtual Directory. Please let me know if you need any additional information.
All the best,
the Telerik team
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