Is there any way an untrusted third party who has access to content from a website over HTTPS can prove the authenticity of the data (i.e. that it was distributed by a server in possession of a specific TLS private key)? The way TLS works makes it such that a packet capture and copy of the master key is insufficient to prove authenticity, since the HMAC key is derived from the master key, which makes it possible to forge the message. Because the third party is untrusted, having them verify the TLS themselves then endorse the authenticity by digitally signing the material is not a solution either.
I'm pretty sure there is no solution under these constraints, but there may be something I missed.