Pseudo-localization conjecture for completed higher Chow cycle complexes

Let $Y$ be a connected quasi-affine $k$-scheme of finite type. Let

$$Y\hookrightarrow X$$

be a closed immersion into an equidimensional smooth $k$-scheme, let $\widehat{X}$ be the completion of $X$ along $Y$, and let $U\subset |Y|$ be a nonempty open subset. Write $\widehat{X}|_U$ for the quasi-affine open formal subscheme $(U,\mathcal{O}_{\widehat{X}}|_U)$, and regard the final $U$ below as the quasi-affine open subscheme $(U,\mathcal{O}_Y|_U)$ of $Y$. Consider the restriction morphisms

$$z^q(\widehat{X},\bullet)\longrightarrow z^q(\widehat{X}|_U,\bullet),\qquad z^q(\widehat{X}\ \operatorname{mod} Y,\bullet)\longrightarrow z^q(\widehat{X}|_U\ \operatorname{mod} U,\bullet).$$

Pseudo-localization conjecture. The respective cokernels $\mathcal{C}_{\bullet}$ and $\mathcal{C}'_{\bullet}$ of these two restriction morphisms are acyclic.

The conjecture is intended to resemble the localization theorem for higher Chow groups and is presented as a possible route toward a pseudo-localization result. The author states that it had not yet been proved in the supplied text; only conjectural consequences are discussed.