Efficient verification of NO answers to NP oracle queries

Ascertain whether there exists a polynomial-time procedure to verify NO instances of NP-oracle queries that arise when checking solutions to TFNP problems with NP-oracle circuits (e.g., PPP^{NP}), without assuming NP = coNP.

Background

When TFNP subclasses are extended with NP oracles (such as PPP^{NP}), verifying solutions may require checking that certain NP queries are NO instances. The authors explicitly note that this verification step is not known to be efficiently doable, which motivates replacing NP oracles with TFNP oracles in their framework.

This unresolved verification issue is a core barrier for keeping such oracle-extended classes within TFNP, as efficient verification of NO instances is tied to coNP-type capabilities.