Speaking of defense-in-depth. I don't understand why do certificates released by public CAs not add an extension to the cert if the cert was given to a DNSSEC v

Speaking of defense-in-depth. I don't understand why do certificates released by public CAs not add an extension to the cert if the cert was given to a DNSSEC validated host. As browsers are unwilling to validate DNSSEC directly it would allow browsers to just check the certificate. This could be complemented with a relevant HTTP header that forces the browser to only accept the certificate if it has been given out to DNSSEC validated host.