Local stability conjecture for finite wreath products with the integers

Let $\Delta$ be a finite group. The wreath product $\Delta \wr \mathbb{Z}$ is locally stable. Finite-wreath-product local stability conjecture. Every finite wreath product of this form should be locally stable. This is presented as a modest first step toward determining whether wreath products of finitely generated amenable locally stable groups are always locally stable; the supplied text gives no resolution.