A pro-2 group with full normal Hausdorff spectra

We construct a 2-generated pro-2 group with full normal Hausdorff spectrum [0, 1] [0,1], with respect to each of the four standard filtration series: the 2-power series, the lower 2-series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro-2 group with full Hausdorff spectrum with respect to the lower 2-series.