Convex recovery conjecture for well-structured sensing matrices

Let $K$ be the number of scatters, let $N_{\mathsf c}$ and $\bar{\mathsf X},\bar{\mathsf Y},\bar{\mathsf Z}$ be the sensing and array parameters, and let $r_{\ell_1}^{\circ}$ denote the rank of the optimal matrix in the atomic-norm semidefinite program. A sensing matrix is well structured as in Definition. Convex recovery conjecture. Theorem should hold for the subset of well-structured sensing matrices provided that

$$K<\min\left\{\frac{N_{\mathsf c}}{2},\max\{\bar{\mathsf X},\bar{\mathsf Y},\bar{\mathsf Z}\}\right\}$$

and

$$r_{\ell_1}^{\circ}<\min\left\{\frac{N_{\mathsf c}}{2},\max\{\bar{\mathsf X},\bar{\mathsf Y},\bar{\mathsf Z}\}\right\}.$$

The conjecture concerns uniqueness of the recovered scatter vector and frequency set through the convex atomic-norm formulation. It is reported as numerically verified, but remains unproved in the supplied text.