The session information that handles authentication should persist through the browser being closed. If clicking on the NiCE logo shows that you are logged in, it makes me wonder if the first page is simply missing a Login Status widget.
I would begin by verifying that the initial page has a Login Status widget. Using the Developer Tools on either Google Chrome or Mozilla Firefox, you should be able to verify that the following Cookies exist under your Portal domain.
If either of these cookies no longer exists after the browser has been closed and re-opened, there may be custom code that is tampering with the session information.
More information on Sitefinity Authentication can be found here: https://www.progress.com/documentation/sitefinity-cms/authentication-and-single-sign-on-(sso)
More information on the Login Status Widget can be found here: https://www.progress.com/documentation/sitefinity-cms/login-status-widget-webforms
Hopefully, some of this information is helpful for you. If you continue to face this issue, please send an email to the support team at email@example.com and we will gladly assist you further with this issue.