Figà-Talamanca–Picardello conjecture on the radicalizer of the Fourier algebra

Let $G$ be an analytic group with compact center and without non-compact nilpotent direct factors. Let $B_0(G)$ denote the Rajchman algebra and define the radicalizer of the Fourier algebra by

$$A_r(G)=\{u\in B(G):\exists k\in\mathbb{N}\text{ such that }u^k\in A(G)\}.$$

Figà-Talamanca–Picardello conjecture. $A_r(G)$ is dense in $B_0(G)$, equivalently

$$\Delta(B_0(G))=G.$$

This conjecture concerns when powers of Fourier–Stieltjes coefficients belong to the Fourier algebra. The supplied text gives negative results when the center is non-compact or the group has a non-compact nilpotent direct factor, while the conjecture asserts density under the stated hypotheses; its resolution is not specified in the source.