Uniform version of the Tame Fontaine–Mazur conjecture

Let ${\rm K}$ be a number field, and let $\Gamma$ be a uniform pro-$p$ group of dimension $d>2$, so that $\Gamma$ is infinite. A Galois extension ${\rm L}/{\rm K}$ is finitely and tamely ramified when it is ramified at only finitely many primes and is tamely ramified at those primes. Uniform Tame Fontaine–Mazur conjecture. There does not exist a finitely and tamely ramified Galois extension ${\rm L}/{\rm K}$ with

$$\operatorname{Gal}({\rm L}/{\rm K})=\Gamma.$$

This is the reformulation of the representation-theoretic conjecture using the existence of uniform open subgroups and the fact that uniform groups of dimension one or two have a quotient isomorphic to ${\mathbb Z}_p$. It remains open in the stated generality.