TITLE: zero-dimensional completely regular space with $\sigma$-complete clopen algebra
QUESTION [6 upvotes]: Suppose $X$ is a zero-dimensional completely regular space (clopen sets form a base) such that the Boolean algebra of clopen sets is a $\sigma$-complete Boolean algebra.  Must $X$ be basically disconnected?  That is, must every cozero set in $X$ have open closure?
I thought this was known, but I can't find it. The point is to show that such $X$ is strongly zero-dimensional, i.e., $\beta(X)$ is zero-dimensional, so that every cozero set is a countable union of clopens.
It should be noted that the analogous statement for extremally disconnected spaces is true: A zero-dimensional space is extremally disconnected (open sets have open closures) if and only if the Boolean algebra of all its clopen sets is a complete Boolean algebra. The proof is an easy exercise.

REPLY [5 votes]: The answer to the posted question is probably no.  Thanks to the above comment by @Joseph Van Name, the claim "yes" is posed as problem 6K in the Porter-Woods book.  Grant Woods has told me in a recent email that this matter came to his attention some time ago, and he concluded that the problem statement in the book should have stated "strongly zero-dimensional" in the assumption. 
A complete answer to the posted question will require a counterexample.  This may not be so well-known.  This will require a zero-dimensional space which is not strongly zero-dimensional, whose Boolean algebra of all clopen sets is $\sigma$-complete.  Perhaps a modification of the Dowker construction would work (see Porter-Woods Problem 4V, or Gillman-Jerison Problem 16M, or Engelking Example 6.2.20).