As discussed on Slack, this would be beneficial as:
1 - RFC 6455 states that WebSocket "is designed to work over HTTP ports 443 and 80 as well as to support HTTP proxies and intermediaries," thus making it compatible with the HTTP protocol. To achieve compatibility, the WebSocket handshake uses the HTTP Upgrade header[1] to change from the HTTP protocol to the WebSocket protocol.
2 - a different port on the same hostname is considered a different origin in CORS policy.
3 - no need to open additional ports through the firewall (especially tricky in corporate environments).