Reiter's sharp decay conjecture for valid functions

Let $\Sigma_n\curvearrowright X_n$ be an asymptotically oligomorphic sequence of group actions, let $G$ be a compact group, and let $\phi\in\widehat G\setminus\\{\mathbf{1}\\}$. For a non-power word $w\in F_r$, write $\operatorname{Ind}_n\phi$ for the induced character of $G\wr_{X_n}\Sigma_n$. Reiter's sharp decay conjecture. The exponent $1/2$ in the bound

$$\operatorname{\mathbb{E}}_w[\operatorname{Ind}_n\phi]=O\left(\dim(\operatorname{Ind}_n\phi)^{-1/2}\right)$$

can be replaced by $1$, and this exponent is tight. The supplied text says the existing theorem gives exponent $1/2$ and that the improvement is conjectural.