When a pro-$p$ group of finite rank can be embedded into the first congruence subgroup of ${\rm GL}_{N}(\mathbb{Z}_{p})$

Question

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ be a odd prime. We say that a pro-$p$ group has finite rank if it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some integer $d$. A pro-$p$ group $G$ is called FAb if $U^{\text{ab}}$ is finite for all open subgroups $U$ of $G$ where $U^{\text{ab}}$ denotes the abelianization of $U$. For example, any open pro-$p$ subgroup of $\SL_n(\mathbb{Z}_p)$ is FAb. We will denote the first congruence subgroup of $\GL_n(\mathbb{Z}_p)$ by $\GL_{n}^{1}(\mathbb{Z}_p)$, i.e. $$\GL_{n}^{1}(\mathbb{Z}_p):=\ker(\GL_n(\mathbb{Z}_p)\to \GL_n(\mathbb{F}_p)).$$

Question: Suppose that $G$ is a torsion-free pro-$p$ group of finite rank.

1. Is there a positive integer $N$ such that $G$ admits an injective homomorphism $G\to \GL_{N}^{1}(\mathbb{Z}_p)$ ?
2. Assume further that $G$ is FAb. Is there an affirmative answer to Question 1? Note that we may assume that $G$ is non-solvable in this case. (Otherwise, it must be trivial.)
3. If the answer to the Question 1 is no, then is it possible to give some sufficient condtions?

Answer 1

The answer to 1 is No. By Corollary 4.5 and Example 4.2 in Analytic pro-p groups of small dimensions, we see that

Every $p$-adic analytic pro-$p$ group of dimension $p$ which embeds into a saturable pro-$p$ group is itself saturable. On the other hand, there exists a torsion-free $p$-adic analytic pro-$p$ group of dimension $p$ which is not saturable.

In other words, any torsion-free $p$-adic analytic pro-$p$ group of dimension $p$ which is not saturable cannot be embeded into a saturable pro-$p$ group, and there really exist torsion-free $p$-adic analytic pro-$p$ group $G$ of dimension $p$ which is not saturable. Since ${\rm GL}_{n}^{1}(\mathbb{Z}_p)$ is saturable, this gives a counterexample to Question 1. However, the group $G$ in the above example is not FAb. So I still don't know the answer to Question 2.