Non-trivial non weakly pseudocompact spaces

A space Z is weakly pseudocompact if Z is Gδ-dense in at least one of its compactifications. In 1996 F.W. Eckertson [3] proposed the following problem: Find examples of Baire non Lindelöf spaces which are not weakly pseudocompact. Eckertson proposed a list of natural candidates. In this article we show that part of this list produces examples of this type by providing examples of product spaces which are Baire non-Lindelöf and not weakly pseudocompact.