Golomb–Moreno conjecture on circular Costas sequences

Let $\mathbb{Z}_n$ denote the cyclic group of order $n$. A circular Costas sequence is equivalent to an injective map $\varphi:\mathbb{Z}_n\to\mathbb{Z}_{n+1}$ whose difference maps

$$\Delta_{\varphi,k}:\mathbb{Z}_n\to\mathbb{Z}_{n+1},\qquad i\mapsto\varphi(i+k)-\varphi(i)$$

are injective for every nonzero $k\in\mathbb{Z}_n$. A Welch sequence is obtained from the Welch construction over a finite field. Golomb–Moreno's conjecture. A Costas sequence is circular if and only if it is Welch. Circular Costas sequences form a stronger class than singly periodic sequences, and the conjecture asserts that the Welch construction gives all of them; its resolution is not established in the supplied text.