Conjecture on asymptotic isometry and asymptotic equivalence

Let $Y$ and $Z$ be unbounded metric spaces, and let $\tilde{r}=(r_n)_{n\in\mathbb{N}}$ be a scaling sequence. For a metric space $W$, write $Seq(W,\tilde{r})$ for its sequence space at scale $\tilde{r}$, and let $Y_1,Z_1\subseteq X$ be unbounded metric subspaces. Asymptotic isometry–equivalence conjecture. The following conditions are equivalent:

1. $Y$ and $Z$ are asymptotically isometric with respect to $\tilde{r}$.
2. There are a metric space $X$ and unbounded subsets $Y_1,Z_1\subseteq X$ such that:
3. $Seq(Y,\tilde{r})$ is pseudoisometric to $Seq(Y_1,\tilde{r})$;
4. $Seq(Z,\tilde{r})$ is pseudoisometric to $Seq(Z_1,\tilde{r})$;
5. $Y_1$ and $Z_1$ are asymptotically equivalent with respect to $\tilde{r}$.

This conjecture describes the proposed interconnection between asymptotically isometric spaces and asymptotically equivalent metric subspaces of a common metric space. The supplied text proves a related implication for asymptotically equivalent subspaces but gives no resolution of the stated equivalence.