N Kuhn has given several conjectures on the special features satisfied
by the singular cohomology of topological spaces with coefficients
in a finite prime field, as modules over the Steenrod algebra. The
so-called realization conjecture was solved in special cases
in [Ann. of Math. 141 (1995) 321–347] and in complete generality
by L Schwartz [Invent. Math. 134 (1998) 211–227]. The more general
strong realization conjecture has been settled at the
prime 2, as a consequence of the work of L Schwartz [Algebr. Geom.
Topol. 1 (2001) 519–548] and the subsequent work of F-X Dehon and
the author [Algebr.
Geom. Topol. 3 (2003) 399–433]. We are here
interested in the even more general unbounded strong realization
conjecture. We prove that it holds at the prime 2 for the class of
spaces whose cohomology has a trivial Bockstein action in high degrees.