When you have page caching turned on in the W3 Total Cache options and a user who is not logged in attempts to rate a review, after refreshing the page, the original cached version of the page appears again instead of the updated page. In order to avoid this you need to add some values to the W3 Total Cache page cache exceptions list.
Within WordPress, go to Performance >> Page Cache >> Advanced and locate the "Never cache the following pages:" option. At the end of the current list add a value like this:
That would be for if you had a minisite called movies. You'll need to add one in the same fashion for each minisite. So for instance in the demo content you would add the following:
Keep in mind these values should match the URL of your single minisite articles. If you are using default permalink structure and your minisite articles are not being served in this manner, you'll need to use a different approach.