Free-by-Demushkin pro-p groups

We give an example of a short exact sequence 1 -> N -> G -> D -> 1 of pro-p groups such that the cohomological dimension cd(G) = 2, G is (topologically) finitely generated, N is a free pro-p group of infinite rank, D is a Demushkin group, for every closed subgroup S of G containing N and any natural number n the inflation map H 2 (S/N, Z/(p(n))) -> H-2(S, Z/(P-n)) is an isomorphism but G is not a free pro-p product of a free pro-p group by a Demushkin group. This is a group theoretic version of a question raised by T. Wurfel for some special Galois groups.