In the book Open Problems in Topology by Jan Van Mill and George M. Reed, the following problem was presented: 108. Is there a para Lindelof Dowker space? Recall that a paraLindelof Dowker space has a locally countable open refinement, satisfies Axiom T4, and is not countably paracompact. Some results on this problem are in http://topology.auburn.edu/tp/reprints/v11/tp11203.pdf, where it is shown that the conditions are preserved under perfect mappings. What is the status of this problem? Any references are appreciated.
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It is now problem 502 in "Open problems in topology II", I would guess it is still open. 

