Membership 2 Pro + IE 11 on Windows 7: Users are logged out automatically

This is quite an urgent problem, as we were planning to launch the site next week, but cannot until this is solved.

We have a site running Membership 2 Pro plugin. Here is the problem we are having when using Internet Explorer 11 on Windows 7 (see below the details of the versions):

1. We log in
2. Try to access any protected content
3. We get the message "content protected" although the user has the permission. Upon further inspection, we can verify that the user is simply not logged in anymore (we can also verify that the wordpress_loggedin cookie has disappeared)

In fact, second step is unnecessary. If you click on any page, we see that we are in fact logged out.

I tried disabling all plugins except Membership Pro 2, and the problem persists. After disabling the Membership Pro 2 plugin, we do not have the problem anymore.

The problem only appears on:
- Windows 7 Professional
- Internet explorer 11 (11.0.9600.17959)
- Plugin version: 4.0.0.5

The plugin does not give the problem on Chrome, Firefox, and even not on IE 11 on Windows 8.