Cardinality of Tukey equivalence classes for Menger and strong Menger pairs in ZFC

Investigate, in ZFC, the number of Tukey inequivalent classes among: (a) pairs (F(M), K(M)) where M is strong Menger; (b) pairs (M, K(M)) where M is Menger; and additionally determine whether 2^c is an upper bound for the number of Tukey inequivalent (M, K(M)) pairs where M is Menger, and whether it is consistent that there are strictly fewer Tukey classes among (F(M), K(M)) with M strong Menger than among (M, K(M)) with M Menger.

Background

Theorem 3.7 shows that, under the hypothesis 2^b > c, there are 2^b-many strong Menger sets with pairwise Tukey-inequivalent pairs (F(M), K(M)). The authors believe such diversity might hold in ZFC, but the hypothesis 2^b > c is currently used.

This question seeks unconditional (ZFC) bounds—both lower and upper—on the number of Tukey classes for compact-cover pairs associated with Menger and strong Menger separable metrizable spaces, and explores potential separations between these two families.