Finite powers of selectively pseudocompact groups

A space X is called selectively pseudocompact if for each sequence (Un)n<ω of pairwise disjoint nonempty open subsets of X there is a sequence (xn)n<ω of points in X such that xn∈Un, for each n<ω, and clX({xn:n<ω})∖(⋃n<ωUn)≠∅. Countably compact spaces are selectively pseudocompact and every selectively pseudocompact space is pseudocompact. We show, under the assumption of CH, that for every positive integer k>2 there exists a topological group whose k-th power is countably compact but its (k+1)-st power is not selectively pseudocompact. This provides a positive answer to a question posed in [10] in any model of ZFC+CH.