The minimal-generator characterisation of uniform pro-p groups

Let $p\geq 3$ and let $G$ be a torsion-free pro-$p$ group of finite rank. Write $d(G)$ for the minimal number of topological generators of $G$, and let $\dim(G)$ denote its dimension as a $p$-adic manifold. A pro-$p$ group is uniform if it is finitely generated, powerful, and torsion-free. Uniformity conjecture. $G$ is uniform if and only if

$$d(G)=\dim(G).$$

Every uniform pro-$p$ group satisfies $d(G)=\dim(G)$, while the converse is the proposed characterisation for torsion-free pro-$p$ groups of finite rank. The conjecture is motivated by the characterisation of $p$-adic analytic groups via finite-rank open pro-$p$ subgroups and by the equality of rank and dimension for torsion-free compact $p$-adic Lie groups when $p\geq 3$.