Successors of (ωω, K(ωω)) in the Tukey order for separable metrizable pairs

Determine whether the pair (ωω, K(ωω)) has any successors in the Tukey order on pairs (M, K(M)) where M is separable metrizable; that is, ascertain whether there exist Tukey equivalence classes strictly above (ωω, K(ωω)) that are successors of it.

Background

The paper describes the initial Tukey structure for pairs (M, K(M)) with M separable metrizable: the minimum class corresponds to compact spaces; its successor corresponds to σ-compact non-compact spaces; and the next class corresponds to analytic non-σ-compact spaces, represented by (ωω, K(ωω)).

Using Theorem 3.4, the authors relate (M, K(M)) ≤T (ωω, K(ωω)) precisely to Menger properties for separable metrizable spaces. Despite partial results and a candidate above (ωω, K(ωω))—namely (K(Q), K(K(Q)))—the existence of successors of (ωω, K(ωω)) remains unsettled.