Finiteness conjecture for circulant complex Hadamard matrices

Let $C_n^{\operatorname{circ}}(\infty)$ denote the set of circulant complex Hadamard matrices of order $n$, with no fixed bound on the orders of their root-of-unity entries, and let $C_n^{\operatorname{circ},1}(\infty)$ be the subset with $1$ on the diagonal. Finiteness conjecture for circulant complex Hadamard matrices. The set $C_n^{\operatorname{circ}}(\infty)$ is finite if and only if

$$n=p_1\cdots p_k,$$

with $p_i$ distinct primes. In addition, in this case, there should be an explicit bound of the form

$$\# C_n^{\operatorname{circ},1}(\infty)\leq 4^n.$$

Earlier results give infinite families when a divisor condition holds and an upper bound for prime order, motivating the proposed characterization of exactly when finiteness occurs; the stated bound remains conjectural.