On the cardinality of almost discretely Lindelof spaces

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