> The latency is the same because modern browsers multiplex HTTP requests over a single TCP connection that is left open.

In my experience this isn’t true; firstly you’re relying on an implementation detail of the platform that you’re executing on, of which you have no control over on the client side. Secondly, even if you aren’t opening a new connection per request, you’re still travelling through an entire HTTP stack implementation rather than the incredibly simple WebSocket protocol - effectively a length and a mask to get the contents, rather than some (in http1 land) fuzzy parser.

If you can guarantee you’re hitting http2 or http3 then you might be closer in latency, but due to the complexity of both I would imagine plain http1 negotiated persistent WebSockets provide the best latency.