Vakil–Wood's homological stability conjecture for symmetric complements

Let $X$ be an irreducible smooth complex variety. For a partition $\lambda$ of $k$, let $W_\lambda(X)$ denote the symmetric complement associated with the stratum indexed by $\lambda$. If $1^j\lambda$ is the partition obtained by adjoining $j$ additional parts equal to $1$, then Vakil–Wood's homological stability conjecture.

$$\dim H_i(W_{1^j \lambda}(X);\mathbb{Q})=\dim H_i(W_{1^{j+1} \lambda}(X);\mathbb{Q})$$

for $j\gg i$. This predicts rational homological stability for symmetric complements as the number of particles increases; the conjecture is presented here without evidence of resolution.