Menger and strong Menger characterizations for Lindelöf spaces via Tukey order

Determine whether, for every Lindelöf space X, the equivalences hold: (a) X is Menger if and only if (X, K(X)) ≥T (ωω, K(ωω)); and (b) X is strong Menger if and only if (F(X), K(X)) ≥T (ωω, K(ωω)).

Background

For separable metrizable spaces M, Theorem 3.4 establishes that (M, K(M)) ≥T (ωω, K(ωω)) if and only if M is Menger, and (F(M), K(M)) ≥T (ωω, K(ωω)) if and only if M is strong Menger. The proof of the forward implication extends to Lindelöf spaces X, but the reverse implication relies on Theorem 3.1, which requires metrizability.

This question asks whether the same Tukey-characterizations of Menger and strong Menger spaces hold in the wider class of Lindelöf spaces, thus generalizing a central result beyond the metrizable setting.