Natural separations among typicality properties

Identify natural Π^1_2-problems that have property P_2 but not property P_1, or property P_3 but not property P_2, for randomness or for genericity.

Background

The paper introduces three increasingly weak notions describing when a problem has rarely solved instances: P_1 requires a computable rarely solved instance, P_2 requires such behavior for a co-small class of oracles relative to instances they compute, and P_3 requires merely the existence of a rarely solved instance. The implications P_1 ⇒ P_2 ⇒ P_3 are shown to be strict for artificially constructed Π^1_2-problems.

For the natural reverse-mathematical principles surveyed in the paper, each problem either has P_1, and hence P_2 and P_3, or fails even P_3. The unresolved question asks whether this apparent dichotomy persists for natural principles, separately under randomness and genericity.