# Is there an $L$-space whose square is selectively $d$-separable?

An $$L$$-space is a hereditarily Lindelof regular space which is not separable.

A space is $$d$$-separable if it contains a dense set which is the countable union of discrete sets.

An $$L$$-space can't be $$d$$-separable, because it has no uncountable discrete subsets, however, by a small modification of his celebrated construction of a ZFC $$L$$-space, Justin Moore provided a ZFC example of an $$L$$-space with a $$d$$-separable square.

Moore, Justin Tatch, An $$L$$ space with a $$d$$-separable square, Topology Appl. 155, No. 4, 304-307 (2008). ZBL1146.54015.

At first glance $$d$$-separability looks like the strongest "separability-type" property you could hope to get in the square of an $$L$$-space, or is it? Consider, for example, the following "selective version" of $$d$$-separability:

A space is called $$D$$-separable if, for every sequence $$\{D_n: n < \omega \}$$ of dense subsets of $$X$$, there are discrete sets $$E_n \subset D_n$$, for every $$n<\omega$$, such that $$\bigcup \{E_n: n < \omega \}$$ is dense.

The above property lies between a property that the square of an $$L$$-space clearly can't have (a $$\sigma$$-disjoint $$\pi$$-base) and a property that the square of an $$L$$-space can have ($$d$$-separability).

QUESTION: Is there an $$L$$-space with a $$D$$-separable square?

For more information about $$D$$-separability see: Bella, Angelo; Matveev, Mikhail; Spadaro, Santi, Variations of selective separability II: Discrete sets and the influence of convergence and maximality, Topology Appl. 159, No. 1, 253-271 (2012). ZBL1239.54014.).