A pro-$p$ group with full normal Hausdorff spectra

For each odd prime $p$, we produce a $2$-generated pro-$p$ group $G$ whose normal Hausdorff spectra $${\mathrm{h}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}_{⊴}^{\mathcal{S}}(G)=\{{\mathrm{h}\mathrm{d}\mathrm{i}\mathrm{m}}_G^{\mathcal{S}}(H)\mid H{⊴}_{\mathrm{c}}G\}$$ with respect to five standard filtration series $\mathcal{S}$ - namely the lower $p$-series, the dimension subgroup series, the $p$-power series, the iterated $p$-power series and the Frattini series - are all equal to the full unit interval $[0,1]$. Here $$\mathrm{h}\mathrm{d}\mathrm{i}\mathrm{m}$_G^{$\mathcal{S}$} \colon { X\mid X \subseteq G } \to[0,1]$ denotes the Hausdorff dimension function associated to the natural translation-invariant metric induced by the filtration series $\mathcal{S}$.