The issue described by the article is only due to the digital signatures occurring in a TLS handshake, not due to the subsequent (symmetric) encryption of the connection (AES-GCM), which as the article explains is already considered quantum safe.

That aside, digital signatures are implemented in terms of asymmetric encryption (with the roles of private and public key reversed), so the quantum safety of asymmetric encryption and of digital signatures, as well as their possible size issues, are really one and the same.