Original text:
an Empty Authenticator, as described in Section 5 of I-D.ietf-tls- exported-authenticator, in a "CERTIFICATE" frame in response to the request, followed by a "USE_CERTIFICATE" frame which references the <del> Empty Authenticator.  In this case, or if the client has not advertised support for HTTP-layer certificates, the server processes the request based solely on the certificate provided during the TLS handshake, if any.  This might result in an error response via HTTP, such as a status code 403 (Not Authorized). </del>  3.

Reviewer comments:
though I'm not sure it's yet precise enough. There are three general classes of errors: The peer did something wrong. There are relatively few of these possible -- indicating more certificates to use than were requested, or more than one unsolicited certificate; sending CERTIFICATE_NEEDED to a peer that didn't indicate support for this extension. The authenticator is corrupt. This might indicate a problem with the TLS connection. The certificate isn't acceptable. There are lots of reasons this might happen. The first two should be at least stream and possibly connection errors. The latter should be handled at the HTTP layer, but previously had error codes defined for them. This PR removes several error codes and recommends handling at the HTTP layer if the certificate isn't accepted.
It is very unclear from the draft how exactly the client should handle a secondary server certificate it rejects. Presumably a bad secondary certificate should not be fatal to streams which don't use it. The draft seems to handwaive at this (though not very clearly, which means uncertainty for the server) in section 4. At the same time, the draft says: This is inconsistent with that idea. Suppose the server pushes some CERTIFICATE frame along with some resources it thinks the client would want. If the client rejects that certificate, this text implies the client must tear down the entire connection, interrupting unrelated requests. A useful heuristic for reasoning through all this would be to consider a client which blindly rejects every secondary certificate. This is especially relevant if the draft does not act on .
The restriction on the server is that it not push unless it has first sent the certificate. If the client rejects the certificate, clearly it will discard the push; the question is whether the client will be able to differentiate failure-to-send from cert-not-accepted. That depends on whether you assume that clients will look at the certificates, even if they aren't accepted; otherwise, I suspect that it can't. However, I think you're right that this is an overreaction. RFC7540 requires that a PUSH_PROMISE for an unproven origin be treated as a stream error, but doesn't specify on which stream. The parent stream still seems excessive, since the server can't know for which origins the client considers it authoritative.

Revised text:
an Empty Authenticator, as described in Section 5 of I-D.ietf-tls- exported-authenticator, in a "CERTIFICATE" frame in response to the request, followed by a "USE_CERTIFICATE" frame which references the  <ins> Empty Authenticator. If the client has not advertised support for HTTP-layer certificates, fails to provide a requested certificate, or provides a certificate the server is unable to verify, the server processes the request based solely on the certificate provided during the TLS handshake, if any.  This might result in an error response via HTTP, such as a status code 403 (Not Authorized). </ins> 3.