Stable Pontryagin-Thom construction for proper maps

We will present proofs for two conjectures stated in arXiv:1808.08073. The first one is that for an arbitrary manifold $W$, the homotopy classes of proper maps $W\times {\mathbb{R}}^n\to {\mathbb{R}}^{k+n}$ stabilise as $n\to \mathrm{\infty }$, and the second one is that in a stable range there is a Pontryagin–Thom type bijection for proper maps $W\times {\mathbb{R}}^n\to {\mathbb{R}}^{k+n}$. The second one actually implies the first one and we shall prove the second one by giving an explicit construction.