The generic polygonal linkage swap-group conjecture

A polygonal linkage is a closed chain of segments with fixed edge lengths, and the swap group $SW_n$ is generated by the elementary swaps $s_i$ acting on its configurations. Let $F_n^0$ be the subgroup of the free group on the swaps with total exponent sum zero, let $R$ be the subgroup acting trivially, and let $Stab$ denote the stabilizer group of a generic linkage. Generic swap-group conjecture. For a generic polygonal linkage, the groups $Stab$ and $R$ coincide; equivalently,

$$SW_n=F_n^0/R=\mathbb Z^{n-1}.$$

The conjecture is motivated by computer experiments showing large, apparently distinct orbits for pentagonal linkages. It asserts that, for generic linkages, the only relations among swap actions are those in $R$, giving a free abelian swap group of rank $n-1$.