Finite generation conjecture for homology of no-k-equal configuration spaces

Let $j$ and $w$ be integers, let $\operatorname{config}(n,w)$ denote the relevant configuration space, and let $\operatorname{FI}_d$ be the category of finite sets and injections with $d=1+\left\lfloor\frac{j}{w-1}\right\rfloor$. The homology groups $H_j(\operatorname{config}(n,w))$ form a sequence indexed by $n$. Finite generation conjecture. For any $j$ and $w$, the homology groups

$$H_j(\operatorname{config}(n,w))$$

form a finitely generated $\operatorname{FI}_d$-module for

$$d=1+\left\lfloor\frac{j}{w-1}\right\rfloor.$$

This would generalize the paper's finite-generation results for disks in a strip and for no-$k$-equal spaces to the spaces $\operatorname{config}(n,w)$ with $w>2$; the source does not provide a resolution of the conjecture.