Comment 6 for bug 1834671

This is implemented in firefox, but not enabled by default indeed. See https://hg.mozilla.org/mozilla-central/rev/1bb8ad865648:

  // Turn off post-handshake authentication for TLS 1.3 by default,
  // until the incompatibility with HTTP/2 is resolved:
  // https://tools.ietf.org/html/draft-davidben-http2-tls13-00
  pref("security.tls.enable_post_handshake_auth", false);

And chrom{e,ium} isn't even considering implementing it until the specification is clarified.