Arithmetic isolation conjecture for Gauss-symbol Hadamard constructions

Let $q\geq3$ be prime, let $K\in M_{M\times N}(\mathbb C)$ be an isolated truncation of $F_p$ with $p$ prime, and let $S,T$ be the sets used in Theorem 4.4. That theorem constructs a partial Hadamard matrix from $K,S,T$. Arithmetic isolation conjecture. The matrix constructed in Theorem 4.4 is isolated, provided that $K$ is an isolated truncation of $F_p$, with $p$ prime, and $S,T$ consist respectively of consecutive odd numbers, and consecutive even numbers. The source notes that the relevant defect computations involve Legendre symbols and roots of unity and leaves the assertion conjectural.