A ZFC example strictly between (ωω, K(ωω)) and K(Q)

Determine whether, in ZFC, there exists a separable metrizable space M such that (ωω, K(ωω)) <T (M, K(M)) <T K(Q), with ωω =T (ωω, K(ωω)).

Background

The authors note that under V = L there is an example (with M = K(N)), while it is consistent that no such M exists if one additionally requires M to be hereditarily Baire. Thus, the existence of a ZFC example, or a consistency result showing nonexistence, remains open.

This question targets the existence of a genuine intermediate Tukey class strictly between (ωω, K(ωω)) and K(Q) in ZFC within the field of separable metrizable spaces.