Full does not imply strong, does it? (Q2577745)

An important open question in the theory of natural dualities [cf. \textit{D. M. Clark} and \textit{B. A. Davey}, Natural dualities for the working algebraist. Cambridge: Cambridge University Press (1998; Zbl 0910.08001)] is whether every full duality is strong. The paper shows that this is not true at the finite level. Since the counterexample is not full it also shows that there is no full duality compactness theorem (although there is a duality compactness one).