The QQR code duality conjecture

Let $p>5$ be an odd prime, let $Q,N\subset GF(p)^\times$ be the quadratic residues and non-residues, and let $C_{NQ}$ be the quasi-quadratic residue code

$$C_{NQ}=\{(r_Nr_S,r_Qr_S):S\subseteq GF(p)\}\subset {\mathbb F}^{2p},$$

where ${\mathbb F}=GF(2)$. The QQR code duality conjecture. If $p\equiv1\pmod4$, then $C_{NQ}\oplus C_{NQ}^{\perp}={\mathbb F}^{2p}$ and $C_{NQ}\cap C_{NQ}^{\perp}=\{\mathbf 0\}$; if $p\equiv3\pmod4$, then $C_{NQ}^{\perp}=C_{NQ}$. The claim is based on computer computations and concerns whether these quasi-quadratic residue codes are complementary or self-dual according to the congruence class of $p$; no proof is supplied.