The custom post type is called Messages and has four additional custom fields on top of the default title and body. These fields store information on the recipient, sender, page from which the conversation was started and details of the first message which enables threaded messages.
In addition to that, its not a live chat system but messaging system.
If you want to bind that to your desired post type instead of Messages then you should have to create all those additional fields with your desired post type.
Here is the sandbox site, you can auto-login to it using the following link:
- hidden link
And you should try to build your message system with your custom post type and add all those additional fields.