Constructive consequences of the remaining Brouwer-code comparison relations

Determine whether each of the two remaining comparison relations among the four interpretations of every Brouwer ordinal code—namely, the comparison from the standard interpretation to the compact interpretation and the comparison from the trichotomous interpretation to the compact totally separated interpretation—implies a constructive taboo.

Background

The paper defines four interpretations of each Brouwer ordinal code: the standard interpretation, the compact interpretation obtained from an extended supremum, the compact totally separated interpretation obtained from an extended sum, and the trichotomous interpretation obtained from a sum indexed by the trichotomous ordinal. Several comparison relations between these interpretations are established under excluded middle, while two corresponding constructive implications are shown to entail limited principles of omniscience: one entails LPO and another entails ¬¬WLPO.

The authors explicitly leave the two remaining comparison implications unresolved and ask whether either or both force a constructive taboo. Resolving this would clarify the constructive strength of the remaining relationships among the ordinal interpretations.