Countable compactness of all higher powers

Determine whether there exists a topological space $X$ such that $X^\omega$ is countably compact but some higher cardinal power $X^\kappa$ is not countably compact.

Background

The paper proves that countable compactness of $X^\omega$ implies bounded cliques on $X$, and discusses examples separating this hypothesis from stronger forms of productive compactness under additional set-theoretic assumptions.

It remains unknown in ZFC whether countable compactness of the countable power can fail to extend to some higher power. Such a space would clarify the relationship between countable productivity and the stronger ultrafilter-compactness conditions considered in the paper.