On my fresh cPanel server I configured the following settings to make things like cpanel.quantumwarp.com, webmail.quantumwarp.com require https, i.e. cPanel webservices access should be HTTPS enforced.

• Home »Server Configuration »Tweak Settings »Redirection
  • Choose the closest matched domain for which that the system has a valid certificate when redirecting from non-SSL to SSL URLs. Formerly known as “Always redirect to SSL/TLS” = On
  • Logout redirection URL = No Redirection

After I did this, I typed into my address bar cpanel.quantumwarp.com (note no protocol, which implies http://) I got the following error:

The page isn't redirecting properly

So  I contacted my server support people and got the following reply

We have tried to set the requiressl and always_redirect_to_ssl for the cPanel proxy domain like cpanel.quantumwarp.com in cPanel configuration, after setting up it cpanel.quantumwarp.com started redirecting to https://cpanel.quantumwarp.com but after that its started showing "Too Many Redirection" error.

Also we tried to force redirect the proxy subdomains to https in apache configuration but no success.
We have now forwarded it to cPanel support team and we will update you once we receive an update from them.

 And here is the response from the cPanel Team

We have received below response from cPanel support team.
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Thank you for your patience as I performed my investigation. The Apache headers module is not installed on your server:
-=-=-=-=-=-=-=-=-=-=-=
[19:11:24 xxxx root@88888888 ~]cPs# rpm -qa | grep mod_headers
[19:11:26 xxxx root@88888888 ~]cPs#
-=-=-=-=-=-=-=-=-=-=-=


The most common issue that occurs when the module is missing is infinite redirect loops when accessing service subdomains. However, it is likely the non-redirection issue is occurring due to the missing module.
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We have now installed the missing apache module and now below links are redirecting to https

Solution

Install mod_headers via EasyApache4