Consider $\mathbb{R}^4$, and note that $\mathbb{R}^4$ is asymptotically diffeomorphic to $\mathbb{R}\times S^3$, meaning that for some compact subset $K$, $\mathbb{R}^4\setminus K$ is diffeomorphic to $\mathbb{R}\times S^3$. For instance, we can let $K$ be a compact ball of radius $R$ around the origin in $\mathbb{R}^4$. Then we have a diffeomorphism $\phi:\mathbb{R}^4\setminus K\to(R,\infty)\times S^3\cong\mathbb{R}\times S^3$, given by $$\phi(x)=(|x|,x/|x|).$$ Clearly, the asymptotic topology of $\mathbb{R}^4$ is not unique, since for instance $\mathbb{R}^4$ is also asymptotically diffeomorphic to $(\mathbb{R}\setminus\{0\})\times S^3$, which can be seen by also letting $K$ include a $3$-sphere with radius larger than $R$. But I suspect that $\mathbb{R}^4$ is not asymptotically homeomorphic to $\mathbb{R}\times S^1\times S^2$, for example. Is this true, and if so, why? Inspired by this, are there any appropriate conditions to require of the set $K$ which makes the asymptotic topology unique up to diffeomorphism or homeomorphism?
Looking at a more general case, let $M$ be a smooth $4$-manifold (I suspect that neither the dimension nor the smooth structure is important here, but I might be wrong) which is asymptotically diffeomorphic to $\mathbb{R}\times S^3$. Can it also be asymptotically diffeomorphic to $\mathbb{R}\times S^1\times S^2$?