On the discretely Lindelof property

A space is discretely property P  if the closure of every discrete subspace has property P. First introduced by Arhangelskii and Tkachuk, who proved independently that spaces that are discretely compact are necessarily compact. They raised the question of whether discretely Lindelof spaces are Lindelof.  Shura proved that they are Linearly Lindelof. 
In joint work with K.P. Hart, we have a couple of interesting (I promise) partial results towards showing that the question may be an independence result.