Existence of a strongly discretely Lindelöf non-Lindelöf space

Establish whether there exists a topological space X such that for every discrete subset D of X the closure of D in X is Lindelöf (i.e., X is strongly discretely Lindelöf) while X itself is not Lindelöf.

Background

A long-standing problem asks whether the property that closures of discrete subsets are Lindelöf can occur in a non-Lindelöf space. The paper contrasts this with the newly introduced SDL property (closures of strongly discrete subsets are Lindelöf), for which a non-Lindelöf example is easy to construct.

Resolving the existence question for the strongly discretely Lindelöf property would clarify the gap between different discrete-set-based Lindelöf-like notions.