It is a standard result that for a CW complex $X$, the chern character

$$\text{ch}: K^*(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^*(X,\mathbb{Q})$$

induces an isomorphism. Suppose now that $X$ is an open manifold and consider the chern character with compact support

$$\text{ch}_{\text{cs}}: K^*_{\text{cs}}(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^{*}_{\text{cs}}(X,\mathbb{Q})$$

Is it still true that it is an isomorphism? It seems to be so for the case of $\text{Tot}(E\to S)$ where $S$ is a compact space and $E$ is a vector bundle. I suppose this is well known, but I couldn't find a reference.