The good old Internet Explorer. It's full with surprises. I just fixed a bug happening only in IE. Everything works just great in every other browser.

The .htaccess file is actually a possible solution for this purpose. The problem with this approach is that the value of AuthUserFile should contain an absolute path. It also works with relative paths, but it is relative based on ServerRoot, which in most of the cases is different from DocumentRoot. I also didn't want to deal with sessions or cookies and after a short research I found the answer.