  In 1983, C. McGibbon and J. Neisendorfer have given a proof for one
conjecture in J.-P. Serre's famous paper (1953). In 1985, another proof was
given by J. Lannes and L. Schwartz.
  Since then, one considers a more general conjecture: if the reduced mod 2
cohomology of any 1-connected polyGEM is of finite type and is not trivial,
then it contains at least one element of infinite height, i.e., non nilpotent.
This conjecture has been verified in several special situations, more
precisely, by Y. Felix, S. Halperin, J.-M. Lemaire and J.-C. Thomas in 1987, by
J. Lannes and L. Schwartz in 1988, and by J. Grodal in 1996.
  In this note, we construct an example, for which this conjecture fails.
