Continuum bound for SDL spaces with a Gδ diagonal

Determine whether every topological space in which the closure of every strongly discrete subset is Lindelöf (an SDL space) and which has a Gδ diagonal necessarily has cardinality at most the continuum.

Background

The paper proves that SDL spaces with a Gδ diagonal of rank 2 have cardinality at most the continuum. It remains unknown whether the rank assumption can be dropped to obtain the same bound for all SDL spaces with a Gδ diagonal.

This question is motivated by classical results (Ginsburg–Woods) for spaces without uncountable closed discrete subsets and by the existence of ccc spaces with a Gδ diagonal and arbitrarily large cardinality, prompting investigation of the SDL condition.