$\DeclareMathOperator\GL{GL}$Let $G$ be a $2$-generator pro-$p$-group of finite rank, i.e. it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some integer $d$. Assume that $G$ is torsion-free. Recall that the dimension $\dim(G)$ of $G$ as a $p$-adic analytic group can be described as $d(U)$ where $U$ is any uniform open pro-$p$-subgroup of $G$ and $d(U)$ denotes the minimal cardinality of a topological generating set for $U$.

**Question:** Is $\dim(G)=2$?