Best lower bound for proof complexity of graph non-automorphism problem 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 proves a graph is rigid ($G \in GNA$)? Also, Is there a plausible conjecture that prohibits sub-exponential certificates for Co-NP-complete problems (analogues to ETH)?
If GNA is in CoNP, and by certificate you mean any string y such that x is in GNA iff xy is accepted by a deterministic turing machine in polynomial time, then a non-trivial lower bound on the certificate size would imply P!=NP. If P=NP, then CoNP=NP=P and there is a deterministic turing machine that accepts GNA without the need of a certificate, or with a certificate whose size is equal to zero.