Conjecture on maximal affine Fourier Hadamard families in dimensions twice a prime

Let $F_d$ be the Fourier matrix in dimension $d$, and for $d=2p$ with $p$ prime let $F_d^{(d/2-1)}(\vec{\xi})$ and its transpose be the affine families defined by Eqs. (mainsol_a)–(mainsol_c), with the phase matrix entries specified by

$$\left(R^{(d/2-1)}(\vec{\xi})\right)_{i,j}=\begin{cases}\bullet & \text{when } i=0 \text{ or } i \text{ is even},\bullet & \text{when } i \text{ is odd and } j=0\pmod{d/2},\xi_{[j-1]} & \text{otherwise.}\end{cases}$$

Maximal-family conjecture. The maximal affine family of complex Hadamard matrices stemming from the Fourier matrix $F_d$ in dimensions $d=2p$, where $p$ is prime, is given by Eqs. (mainsol_a) to (mainsol_c). The paper notes agreement with the maximal affine families in the catalog for $d=2,4,6,10,14$, but provides no resolution for the stated general conjecture.