Finiteness conjecture for top integral cohomology of configuration spaces

Let $M$ be an orientable even dimensional manifold. Assume

$$\operatorname{rank}(H^{d-1}(M;\mathbb{Z}))=0.$$

Finiteness conjecture. There exist $k_{0}\in\mathbb{N}$ such that

$$H^{\operatorname{top}}(C_{k}(M);\mathbb{Z})$$

is finite for every $k\geq k_{0}$.

This conjecture concerns the eventual finiteness of the top nonzero integral cohomology groups of configuration spaces when the $(d-1)$st cohomology of the manifold has rank zero. The preceding examples indicate finiteness for configuration spaces of $\mathbb{R}^{2}$, while the general pattern in this setting remains unclear.