Let $K$ be an algebraic number field and let $G$ be either the Galois group of the maximal algebraic extension of $K$ in a separable algebraic closure $\overline{K}$ unramified outside a finite sent of non-archimedean places of $K$, or the Galois group of a the completion of $K$ at some place. Let $R$ be some simple non-archimedean field or ring such as $\mathbb{F}_p$,  $\mathbb{Z}_p$, or $\mathbb{Q}_p$. Is anything known about lifting continuous homomorphisms $G \to \operatorname{GL}_n(R[[x_1, \dots, x_k]])$ to continuous homomorphisms $G \to \operatorname{GL}_n(R\langle\frac{x_1}{r_1}, \dots, \frac{x_k}{r_k}\rangle)$? In other words, does there exist positive real numbers $r_1, \dots, r_k$ such that that there is a lift in general? If not, is there some criteria known for their existence? The notation $R\langle\frac{x_1}{r_1}, \dots, \frac{x_k}{r_k}\rangle$ refers to a Tate algebra.