A space with normal countable-to-one regular preimages

It is known that a space with every countable-to-one regular preimage normal must be at least ⩽ scc-cwN. We construct a model of ZFC + CH with a normal space X that has every countable-to-one regular preimage normal but is not ⩽ scc+-cwH, so that ZFC does not imply stronger separation properties for these spaces. X is also an example of a Moore space that is ⩽ ℵ1-paracompact but not ⩽ ℵ2-paracompact.