Existence of a TFΣ3^P problem not collapsing to TFNP with a Σ2^P oracle; further collapses for Empty and Short-Choice

Determine whether there exists a total search problem in TFΣ3^P that does not collapse to any TFNP subclass when given access to a Σ2^P oracle; additionally, ascertain whether the TFΣ2^P problems Empty and Short-Choice further collapse to smaller TFNP subclasses relative to appropriate oracles.

Background

The paper develops a general framework showing collapses of problems in the total function hierarchy TFΣi^P to TFNP subclasses equipped with Σ{i-1}^P oracles. Several prominent TFΣ3^P problems (e.g., King in this work; Shattering in prior work) collapse to smaller classes with Σ2^P access.

This raises the question of whether any TFΣ3^P problem resists such collapses. For TFΣ2^P, only Empty and Short-Choice are highlighted as not yet known to collapse further. Understanding these boundaries would clarify the landscape of the total function hierarchy under oracle access.