Graph automorphism problem ( GA) of determining whether a graph has a nontrivial automorphism is a good candidate for a problem in $NP$-intermediate. I'm looking for references that study the certificate complexity of graph non-automorphism (GNA= {G| G is rigid or asymmetric graph}).

What is the best known lower bound on the length of certificates that prove a graph is rigid ($G \in GNA$)? Also, Is there a plausible conjecture that prohibits sub-exponential certificates for $CoNP$-complete problems (analogues to ETH)?