Browsers try to reuse the same TLS session for multiple TCP connections to the same server. But if this is possible depends on the server: if the server does not support session reuse a full handshake need to be done for each new TLS connection.
Edit: This answer cuts across multiple issues without answering the question posed. It is true that browsers use "Connection: Keep alive" to keep sockets open to reduce overhead. It's also true that browsers use "socket pooling" to minimize the number of concurrent connections to a given server, and that TLS uses a Session-Id to avoid repeating the initial handshake that sets up a secure session.
It does not follow from any of this, however, that the multiple sockets a browser uses in concurrent TLS connections to the same server use the same Session keys. A vendor used WireShark to monitor such a connection for me after I posted this question and reports that each of the 6 sockets Chrome opened to the same server used different Session Ids, which suggests that Chrome may not.
But I'm searching for a definitive answer, preferably in the HTTP standard.
Thanks for your answer, nonetheless.