Is there a space $X$ that is not homotopy equivalent to a finite-dimensional CW complex for which there exists a space $Y$ such that the product space $X\times Y$ is homotopy equivalent to a finite-dimensional CW complex? If so, how might we construct an example?

A first consideration could be where $X$ has infinitely many nontrivial homology groups, in which case $X$ is not homotopy equivalent to a finite-dimensional CW complex. Yet, in this case it follows from the Künneth Formula that no suitable $Y$ exists. Of course, this only rules out spaces with infinitely many nontrivial homology groups.

Another consideration could be where $X$ is an Eilenberg–Maclane space for which $\pi_1(X)$ has non-trivial torsion, in which case $X$ is, again, not homotopy equivalent to a finite-dimensional CW complex. In this case, if $\pi_1(X)$ is abelian then $X$ is homotopy equivalent to $L\times Z$ for some other space Z and an infinite-dimensional lens space $L$, and hence $X$ has infinitely many nontrivial homology groups, which implies, again, that no suitable $Y$ exists. This has led me to wonder if there exists a space $Y$ and a noncommutative group $G$ with non-trivial torsion such that $Y\times K(G,1)$ is homotopy equivalent to a finite-dimensional CW complex, where $ K(G,1)$ denotes an Eilenberg–Maclane space.

Any insight into approaching this problem is greatly appreciated.