Cardinality bound |X| ≤ 2^{χ(X)} for Hausdorff SDL spaces

Ascertain whether every Hausdorff space X in which the closure of every strongly discrete subset is Lindelöf (an SDL space) satisfies |X| ≤ 2^{χ(X)}.

Background

The authors prove that Urysohn SDL spaces satisfy |X| ≤ 2^{χ(X)} via bounds involving the weak Lindelöf degree of closed sets. It remains unclear whether the same cardinality bound holds under the weaker Hausdorff separation axiom without Urysohn.

This question seeks to extend the established Urysohn result to the broader Hausdorff class of SDL spaces.