Galois descent conjecture for topological restriction homology

Let $K$ be a field with valuation ring $V$, residue field $k$, and let $\bar{K}$ be an algebraic closure of $K$ with Galois group $G_K$. Let $\bar{V}$ be the integral closure of $V$ in $\bar{K}$. Assume that $k$ is perfect. Galois descent conjecture. For all $q>0$, the canonical map

$$\operatorname{TR}_q^{\,\boldsymbol{\cdot}}(V|K;p,\mathbb{Q}_p/\mathbb{Z}_p) \to \operatorname{TR}_q^{\,\boldsymbol{\cdot}}(\bar{V}|\bar{K};p,\mathbb{Q}_p/\mathbb{Z}_p)^{G_K}$$

is an isomorphism of pro-abelian groups, and

$$H_{\operatorname{cont}}^i(G_K,\operatorname{TR}_q^n(\bar{V}|\bar{K};p,\mathbb{Q}_p/\mathbb{Z}_p))=0$$

for all higher continuous cohomological degrees $i$. This predicts Galois descent for the relevant topological restriction homology groups and the vanishing of the higher obstruction groups; its resolution is not specified in the source.