You can definitely setup the CSS to perform a URI keep-alive over HTTPS.
keepalive type http encrypt
However, in order for this configuration to work properly your CSS must contain the SSL module as the service will need to be setup as a "type ssl-accel-backend". This will allow the CSS to encrypt the keep-alive request and decrypt the servers response using the cert/keys defined within the backend-server configuration within the ssl-proxy-list.
Does your CSS contain an SSL module?
- Jason