Hesselholt's Galois descent conjecture for topological cyclic homology

Let $K$ be a local field with algebraic closure $\bar{K}$, let $V$ be its valuation ring, and let $\bar{V}$ be the integral closure of $V$ in $\bar{K}$. Write $G_K=\operatorname{Gal}(\bar{K}/K)$, and let $\operatorname{TR}_q^{\boldsymbol{\cdot}}(-|-;p,\mathbb{Q}_p/\mathbb{Z}_p)$ denote the resulting pro-abelian groups with their continuous $G_K$-action. For positive integers $q$, consider the canonical map

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

Hesselholt's descent conjecture. For all positive integers $q$, this canonical map is an isomorphism of pro-abelian groups, and the higher continuous cohomology groups of the pro-$G_K$-module $\operatorname{TR}_q^{\boldsymbol{\cdot}}(\bar{V}|\bar{K};p,\mathbb{Q}_p/\mathbb{Z}_p)$ vanish.

This is a Galois descent statement for the positive-degree topological restriction homology of a local field and its algebraic closure. The source says that the conjecture was first formulated in Hesselholt's earlier work; no resolution is supplied here.