D-space consequences for Lindelöf spaces Tukey-below K(Q) or K(M)

Determine whether every Lindelöf space X with (X, K(X)) ≤T K(Q) is a D-space; and determine whether X is a D-space whenever (X, K(X)) ≤T K(M) for some separable metrizable space M.

Background

Menger spaces are known to be D-spaces. Since Theorem 3.4 ties Menger properties to Tukey-below (ωω, K(ωω)) for separable metrizable spaces, it is natural to ask whether similar D-space conclusions follow when a Lindelöf space is Tukey-below K(Q) or, more generally, K(M) for separable metrizable M.

Affirmative answers would extend D-space results beyond classical Menger settings using the language of compact-cover Tukey structure.