On the cardinality of almost discretely Lindelof spaces

A space is said to be "almost discretely Lindelöf" if every discrete subset can be covered by a Lindelöf subspace. Juhász, Tkachuk and Wilson asked whether every almost discretely Lindelöf first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under 2<= (which is a consequence of Martin's Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhász, Soukup and Szentmiklóssy. We conclude with a few related results and questions.