Completeness conjecture for the sequence pseudometric

Let $(X,d)$ be an unbounded metric space and let $\tilde{r}=(r_n)_{n\in\mathbb{N}}$ be a scaling sequence. Define

$$d^{\tilde{r}}(\tilde{x},\tilde{y})=\limsup_{n\to\infty}\frac{d(x_n,y_n)}{r_n}$$

for $\tilde{x},\tilde{y}\in Seq(X,\tilde{r})$; this is a pseudometric on $Seq(X,\tilde{r})$. Completeness conjecture. The pseudometric space $(Seq(X,\tilde{r}),d^{\tilde{r}})$ is complete for every unbounded metric space and every scaling sequence $\tilde{r}$. The claim concerns completeness of the sequence pseudometric introduced for studying metric spaces at infinity; the supplied text gives no evidence that it has been proved or disproved.