I've been tasked with making all the sites on our SharePoint Servers require SSL connections. I've successfully bound our SharePoint Web Applications to https and enabled Require SSL for those web apps in IIS. Those work perfectly, as long as I do not require SSL for the SharePoint Web Services site.
However, when I try to enable Require SSL for SharePoint Web Services, the Security Token Service stops functioning and becomes unable to issue security tokens. Some User Profile jobs also start failing. SharePoint Web Services does have an https binding using port 32844, and I've verified that I can connect to that port. However, when I enable Require SSL then check the SharePoint logs, I see that the Security Token Service Application is continuing to try to connect to the HTTP endpoint on port 32843. I've tried editing the web.config file to see if I can force it to use HTTPS instead, but it always uses HTTP.
Am I stuck with HTTP? Can Require SSL be enabled for SharePoint Web Services, or is this option simply not supported?