Regular complex Hadamard matrices of order seven

Let $F_7$ be the Fourier matrix of order $7$, and let $P_7^q$ denote the displayed Petrescu family, with $q\in\mathbb T$. A regular complex Hadamard matrix is one whose row scalar products decompose as sums of cycles. Order-seven classification conjecture. $F_7,P_7^q$ are the only regular complex Hadamard matrices at $N=7$. The source reports computational and number-theoretic evidence, but does not establish the classification.