Density conjecture for annihilators of ${}_{\Sigma}L_p$

Let $\Sigma$ be a smooth manifold with the $L_p$ restriction property, and let $\Pi_{\Sigma}$ denote the restriction map to $\Sigma$. Write $\operatorname{Ann}_{L_q}({}_{\Sigma}L_p)$ for the annihilator of ${}_{\Sigma}L_p$ in $L_q$. Density conjecture. The set

$$\left\{g\in L_q(\mathbb{R}^d)\mid \exists\,\zeta\in C^{\infty}(\Sigma)\text{ such that }\widehat g=\Pi_{\Sigma}^*[\zeta]\right\}$$

is dense in $\operatorname{Ann}_{L_q}({}_{\Sigma}L_p)$. The conjecture was proved in the source for $\Sigma=\mathrm{S}^{d-1}$ and similarly for more general quadratic surfaces; the general manifold case remains open in the supplied text.