The torsion-free rank of homology in towers of soluble pro-p groups

We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{$\mathbb{Q}$_p} (H_1(M, $\mathbb{Z}$p) \otimes{$\mathbb{Z}$_p} $\mathbb{Q}$_p )$,as$M$runsthroughallpro-$p$subgroupsoffiniteindexin$G$.