The formally self-dual extension conjecture for QQR codes

Let $p>5$ be an odd prime, let $C_{NQ}\subset {\mathbb F}^{2p}$ be the quasi-quadratic residue code, and let $C'$ be the code spanned by $C_{NQ}$ and the all-ones codeword. Let $A=[A_0,A_1,\ldots,A_n]$ be the weight distribution vector of $C_{NQ}$, and define $A^*=[A_n,A_{n-1},\ldots,A_0]$. The formally self-dual extension conjecture. If $p\equiv1\pmod4$, then $C'$ is formally self-dual, has dimension $p$, and its weight distribution vector is $A+A^*$. The claim is presented as supported by computer computations, with no proof supplied.