HTTP keepalive should get you much of the same benefit, though? The major improvement comes from fewer headers sent over websockets?

SSL Session Reuse needs to be configured as well, otherwise every single request will have to perform a new handshake, independently of keepalive.

In practice I've always observed HTTPS requests to take at least twice as long compared to a back and forth through a websocket, even when SSR and keepalive are enabled. I don't know if having both theorically allows for single-roundtrip HTTPS requests.