User Login too persistent?

Not sure if my problem is with my NTLM authentication or with the webserver_auth module, but here goes:

My config:

• Apache 2 on CentOS
• Using Apache2::AuthenNTLM as described here.
• Fresh 4.7.5 Drupal install with webserver_auth module

What works:

1. When a user attempts to loads a page, the apache server first prompts them for login credentials, authenticates them with the domain contoller, and attempts to serve the page.
2. Then Drupal calls webserver_auth_init, and checks their webserver login status, sees they are logged in, and logs the user into drupal (registering a new user, if necessary)
3. Each subsequent page load seems to call webserver_auth_init, and since the user is logged into drupal, the page is served.

My problem:

1. Suppose that user #1 closes their browser, terminating the apache ntlm session.
2. If user #2 opens the browser on that compter and goes to the drupal site, apache will prompt them for their userid & password, authenticate them with the domain contoller, and let drupal serve the page.
3. But when drupal calls webserver_auth_init, it thinks that user #1 is still logged into drupal and allows user #2 to browse the site logged in as user #1

Do I need to do something to get drupal to retrieve the user info from apache for each page load?