It is well-known that if we are given two smooth manifolds (without) boundary, whose underlying topological spaces are homotopic, then the de Rham cohomologies $H^k_{dR}$ of $M$ and $N$ are isomorphic for all $k\geq 0$ [See Theorem 17.11 in Lee's smooth manifolds book]. I was wondering if a similar result holds for the compactly supported de Rham cohomologies $H^k_c(M)$ and $H^k_c(N)$. I know that such a result doesn't hold if we only demand a standard homotopy relation. However Lee mentions shortly in his book that proper smooth maps induce maps between the compactly supported groups. In Exercise 6-8 of Lee's book we are asked to show that every proper continuous map is homotopic to a proper smooth map, but I'm not sure if the homotopy is in fact proper, which according to this thread Invariance of de Rham cohomology with compact support and the cited book by Michor seems to be what we need in order to establish the result. Strangely enough I couldn't find any reference which explicitly states that the compactly supported de Rham cohomology groups are isomorphic for homeomorphic manifolds.
To explicitly state my question: If the underlying topological spaces of two smooth manifolds (without boundary) $M$ and $N$ are homeomorphic, are their compactly supported de Rham cohomologies isomorphic?
Kind regards