  We construct a $2$-generated pro-$2$ group with full normal Hausdorff
spectrum $[0,1]$, 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.
