Yes, this works fully with an Apache reverse proxy. This is my Apache config to make it all work:
<VirtualHost *:443>
ServerName esxi.example.com
ServerAlias esxi
SSLEngine On
SSLCertificateFile /etc/ssl/certs/example.crt
SSLCertificateKeyFile /etc/ssl/private/example.key
SSLCertificateChainFile /etc/ssl/certs/sub.class2.server.ca.pem
SSLProxyEngine On
SSLProxyCACertificateFile /etc/ssl/certs/example.crt
ProxyRequests Off
ProxyPreserveHost On
RewriteEngine on
ProxyPass /sdk/ https://esxi/sdk/
ProxyPassReverse /sdk/ https://esxi/sdk/
ProxyPass /ticket/ wss://esxi/ticket/
ProxyPassReverse /ticket/ wss://esxi/ticket/
ProxyPassMatch ^(/screen.*)$ https://esxi.example.com$1
ProxyPass / https://esxi/ui/ retry=1 keepalive=On
ProxyPassReverse / https://esxi/ui/
RequestHeader set X-Forwarded-HTTPS "0"
CustomLog ${APACHE_LOG_DIR}/access_vmware.log combined
</VirtualHost>