Generating-set invariance conjecture for forcing properties

Let $\Gamma$ be a finitely generated, residually finite group, and let $S_i$ for $i=1,2$ be finite generating sets with $\operatorname{Id}\notin S_i$ and $S_i^{-1}=S_i$. Set

$$F_i=\operatorname{Cay}(\Gamma,S_i).$$

Let $P$ be one of the properties of being connectivity-forcing, being expansion-forcing, or forcing a giant component. Generating-set invariance conjecture. The pair $(F_1,r)$ has property $P$ for all sufficiently large $r$ if and only if $(F_2,r)$ has property $P$ for all sufficiently large $r$. This predicts that, for residually finite groups, each of the three forcing properties is independent of the chosen finite symmetric generating set once the radius is sufficiently large.