(This is a re-post of 1)

In the paper "On the realizability of singular cohomology groups" by Kan and Whitehead, it is shown that there is no space $X$ and integer $n\geq 1$ such that $H^{n−1}(X)=0$ and $H^n(X)=\mathbb{Q}$ (cohomology with integral coefficients).

At the very end of the article there is a remark where it is stated that, at the time of writing (around 1960, I suppose), it was not known whether $\mathbb{Q}$ could be a (integral) singular cohomology group at all.

My question is: is this still not known?