  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 $n\to\infty$, 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.
