In at least one place (src/js/modal.js#L1054), Convenient Discussions prepends "https:" to the value of $wgServer.
However, if $wgServer specifies a URL with a protocol (e.g. https://en.wikipedia.org), the URL that Convenient Discussions obtains will start with https:https://en.wikipedia.org.
Convenient Discussions should probably check whether $wgServer is a protocol-relative or a protocol-absolute URL, and prepend "https:" accordingly.