Truncation conjecture for prime Fourier matrices

Let $F_{S,p}$ denote the partial Fourier matrix associated to a subset $S\subset\mathbb Z_p$, where $p$ is prime. Truncation conjecture. There exists a constant $\varepsilon>0$ such that $F_{S,p}$ is isolated, for any $p$ prime, once $S\subset\mathbb Z_p$ satisfies $|S|\geq(1-\varepsilon)p$. This proposes isolation for sufficiently large truncations of prime-order Fourier matrices, a rectangular analogue of the known isolation result for $F_p$.