Squares of Menger-bounded groups

Using the Continuum Hyporthesis, we prove that there is a Menger-bounded (also called o-bounded) subgroup of the Baer-Specker group Z^N, whose square is not Menger-bounded. This settles a major open problem concerning boundedness notions for groups, and implies that Menger-bounded groups need not be Scheepers-bounded. This also answers some questions of Banakh, Nickolas, and Sanchis.