I’ve found that the issue comes from a normalization function the plugin performs. There are a couple of ways around this.
1) The easiest way to get around it is to change your Site Address value on your main settings page in WordPress so that it is https. That will circumvent the part of the plugin that is changing them back to http.
2) The other way is to open the file “/lib/WordPressHTTPS/Module/Parser.php” and comment out the second conditional in the function normalizeElements(). That will prevent the normalization from happening at all.
I’ve emailed the plugin author, but no response, so I’m not sure when a fix might make it into the plugin.