Non-compact three-manifolds with the same proper homotopy type are homeomorphic?

Question

I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):

Let $M, M'$ be two non-compact connected $3$-manifolds with the same proper homotopy type. Then $M$ is homeomorphic to $M'$.

Recall that two spaces $X$ and $Y$ are said to have the same proper homotopy type if there are proper maps $f\colon X\to Y$ and $g\colon Y\to X$ such that both $f\circ g$ and $g\circ f$ are properly homotopic to the identity maps. A proper homotopy from $X$ to $Y$ is a homotopy, that is, a map $H\colon X\times [0,1]\to Y$, which is a proper map.

Answer 1

Take a look at the example shown at Remark 5.9 b) in the paper "A topological equivalence relation for finitely presented groups."by M. Cárdenas, F.F. Lasheras, A.Quintero and R.Roy. Journal of Pure and Applied Algebra, DOI 10.1016/j.jpaa.2019.106300 . The numerably punctured spaces $\mathbb R^3$ and the "semispace" $\mathbb R^3_+$ are properly homotopic but not homeomorphic. (It works for $n\geq 2$).