Facebook Connect looks like it hasn't worked when you are redirected back to your site

With a fresh install of Drupal 7 and fb-7.x-4.x-dev, I've enabled fb and fb_connected and added an application to the site.

As an anonymous user, if I visit the homepage, then click Facebook Connect, authenticate with Facebook and Authorize the app, I am eventually redirected back to my site's homepage. However, the "Facebook Connect" link is still visible. I would expect this to be replaced with my name.

If I press the browser's reload button, the Facebook Connect link is gone and my name is displayed instead.

This is quite confusing to users, as when they return to the site it gives the impression that it hasn't worked. Can we have their name display instead of Facebook Connect immediately on arriving back at the site, without having to press refresh?

I've created a quick video to show what I mean, incase it's not clear from the above:
https://www.youtube.com/watch?v=qH32rQumvfw&feature=youtu.be