Perfectly normal but not collectionwise normal space in ZFC

Question

In the article A Perfectly Normal, Locally Compact, Noncollectionwise Normal Space Form $\lozenge^\ast$ by Daniels and Gruenhage (I presume "form" is a typo and it should be "from", but it's like that in the title) they show that under $\lozenge^\ast$ there exists a perfectly normal, locally compact, non-metrizable Moore space, in particular not collectionwise normal.

In the same article they mention that existence of perfectly normal, locally compact, not collectionwise normal space is independent of ZFC.

Also they mention that Fleissner space constructed under CH of the article Normal nonmetrizable Moore space from continuum hypothesis or nonexistence of inner models with measurable cardinals is perfectly normal, although I am not sure if we need to assume $\lozenge^\ast$ for it to be $T_6$ or not.

Research seems to revolve around perfectly normal, locally compact, non-collectionwise normal spaces, I am looking for a space which isn't locally compact.

Bing's discrete extension space is an example of a $T_5$ but not collectionwise normal space.

Can ZFC can prove that there exists a $T_6$ space which isn't collectionwise normal?

Answer 1

Bing's Example H is found in

R. Bing, Metrization of Topological Spaces, Canadian J. Math. 3 (1951), 175-186.

It is on the page following his more famous Example G. It is constructed in ZFC and has the following properties:

"EXAMPLE H. A normal topological space that is not collectionwise normal and in which each closed set is an inner limiting ($G_\delta$) set."

There is some anayslis of this space on Dan Ma's blog he shows that Example H is not even collectionwise Hausdorff. Furthermore, the space is subparacompact, but not metacompact.