On your site the user is actually authenticated properly. However the page that is loaded next is coming from cache. If you refresh the page, you'll see the logged in user. Please check whether the System cache plugin is enabled (which should not be enabled when authentication is needed) or the Global Configuration Settings for caching.
The developer tools in Chrome show a status code of 200 (from disk cache).
If you'd like a more detailed explanation about why caching could cause this, check out the
explanation at the bottom of this thread
-Melissa