While most of what Travis said is factually true, I'm unclear why he answered "no" to your question.
When the client opens the HTTPS connection to the webserver, the webserver may serve up JS that initiates WebSocket connections using wss:// to a different origin (may be the same host, but as you described, it is a different port). This is a different connection. An SSL handshake has to be separately performed for each connection.
What Travis said is true for a SSL session, and if the WebSocket used the same connection as the https:// document fetch, then it would be correct to answer "no". But since the wss:// WebSocket connection initiation will use a different connection, it requires another SSL handshake.