Extremal configuration-space homology quasi-polynomial degree conjecture

For an orientable manifold $M$ of even dimension $d\geq 2$, let

$$F_n(M):=\{(x_1,\ldots,x_n)\in M^n\mid x_i\neq x_j\text{ for }i\neq j\}$$

be the ordered configuration space, let $B_n(M):=F_n(M)/S_n$ be the unordered configuration space, and set $\nu_n=(d-1)n+1$. Write $Q_M^{\nu_n-i}(n)=\dim H_{\nu_n-i}(B_n(M);\mathbb{Q})$ for the eventual quasi-polynomial describing the extremal homology in degree $\nu_n-i$.

Extremal quasi-polynomial degree conjecture. If $H_{d-1}(M;\mathbb{Q})$ and $Q_M^{\nu_n-i}(n)$ are non-trivial, then the degree of the quasi-polynomial $Q_M^{\nu_n-i}(n)$ is

$$\dim H_{d-1}(M;\mathbb{Q})-1$$

for $i\geq 0$.

This extends the known degree formulas for the top extremal homology of closed orientable even-dimensional manifolds and for the next extremal group when the manifold is not closed. The conjecture concerns the degree of every non-trivial extremal homology quasi-polynomial, while its resolution is not specified in the source.