No, this only takes care that you don't have to place the https part of a site in the private_html and that you can use http as wel as https in public_html.
They are using some kind of redirect, maybe by a redirect made in their customer panel, but they did it wrongly, should have used a 301 redirect:
AWD is right, you should use a .htaccess or vhost redirect.
This is what I have had ....
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# Https redirect
RewriteEngine On
RewriteCond %{HTTPS} off
RewriteRule ^(.*)$ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301]
# Announcements
RewriteRule ^announcements/([0-9]+)/[a-z0-9_-]+\.html$ ./announcements.php?id=$1 [L,NC]
RewriteRule ^announcements$ ./announcements.php [L,NC]
# Downloads
RewriteRule ^downloads/([0-9]+)/([^/]*)$ ./downloads.php?action=displaycat&catid=$1 [L,NC]
RewriteRule ^downloads$ ./downloads.php [L,NC]
# Knowledgebase
RewriteRule ^knowledgebase/([0-9]+)/[a-z0-9_-]+\.html$ ./knowledgebase.php?action=displayarticle&id=$1 [L,NC]
RewriteRule ^knowledgebase/([0-9]+)/([^/]*)$ ./knowledgebase.php?action=displaycat&catid=$1 [L,NC]
RewriteRule ^knowledgebase$ ./knowledgebase.php [L,NC]
# OpenID Discovery Document (
http://openid.net/specs/openid-connect-discovery-1_0.html)
RewriteRule ^.well-known/openid-configuration ./oauth/openid-configuration.php [L,NC]
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Does not work... this is why my original post...