Conjecture F on homological stability for complements of discriminants

Let $X$ be an irreducible smooth complex variety, let $^j\lambda$ denote the partition obtained by adjoining $j$ parts equal to $1$ to a partition $\lambda$, and let $W_{\lambda}(X)$ be the complement in $\operatorname{Sym}_k(X)$ of the closure of the stratum associated with $\lambda$. Conjecture F. For $j$ sufficiently large relative to $i$,

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

The paper proves this conjecture and strengthens it to an isomorphism for arbitrary connected manifolds of dimension at least $2$, with an explicit stability range; therefore the conjecture is resolved.