Spectrally reasonable measures conjecture for locally compact Abelian groups

Let $G$ be a locally compact non-discrete Abelian group. Write $4\mathcal{S}(G)4$ for the spectrally reasonable measures on $G$, $\mathcal{M}_{00}(G)$ for the measures with Rajchman-type vanishing Fourier–Stieltjes transform, and $\operatorname{lin}(\delta_0)$ for the linear span of the point mass at the identity.

Spectrally reasonable measures conjecture.

$$\begin{cases} \mathcal{S}(G)=\mathcal{M}_{00}(G)\oplus\operatorname{lin}(\delta_0), & \text{if }G\text{ is compact},\\ \mathcal{S}(G)=\operatorname{lin}(\delta_0), & \text{if }G\text{ is non-compact}. \end{cases}$$

This conjecture proposes a complete description of spectrally reasonable measures for all locally compact non-discrete Abelian groups. The surrounding discussion notes that the compact case is supported by the results proved earlier, while the argument does not extend directly to all compact groups, including groups of $p$-adic integers; the general assertion remains open.