Existence conjecture for asymptotically Euclidean isometric immersions

Let $(M^n,g)$, with $n 3$, be an asymptotically Euclidean Riemannian manifold of order $\tau>0$. An immersion $\psi\colon M\to\mathbb{R}^d$ is asymptotically Euclidean of order $\tau$ when it satisfies the asymptotic conditions in the paper's Definition 1.

Asymptotically Euclidean immersion conjecture. There exists an asymptotically Euclidean isometric immersion

$$\psi\colon M\to\mathbb{R}^d$$

of order $\tau$.

This is motivated by Nash's isometric immersion theorem, but the asserted preservation of the asymptotically Euclidean structure is not established in the source.