Isometry between the asymptotic completion and the free-group completion

Restrict to the free group $G=F(A)$, let $\Theta\subseteq P(G)$ be a set of cyclically reduced representatives for the equivalence classes in $P(G)$, and let $\tau$ be the stable norm. Write $\hat{\mathcal C}(G)$ for the relevant asymptotic completion and $\hat{\mathcal D}(\Theta;\tau)$ for the completion associated with $\Theta$ and $\tau$. Isometry conjecture. With this notation, $\hat{\mathcal C}(G)$ is isometric to $\hat{\mathcal D}(\Theta;\tau)$. Since the inclusion map into the completion has dense image, the conjecture asserts that the resulting completion does not add any metric discrepancy; the source gives no resolution.