TITLE: Zero-dimensional $F$-space which is not strongly zero-dimensional
QUESTION [7 upvotes]: Does anyone know of an example of a (Tychonoff) $F$-space which is zero-dimensional but not strongly zero-dimensional?
By an $F$-space we mean every cozeroset is $C^*$-embedded.
By zero-dimensional we mean has a base of clopen sets.
By strongly zero-dimensional we mean every cozeroset is a countable union of clopen sets.

REPLY [9 votes]: Alan Dow and I constructed an example.
It can be found by following this link.
The construction is based on the locally compact modification of Dowker's example in this answer.
The final version of the example is now available on Arxiv.org and in Topology and its Applications