Golomb–Taylor conjecture on singly periodic Costas sequences

Let a Costas sequence be a sequence whose difference triangle has no repeated entry in any row. A sequence is singly periodical when every circular shift of its terms is also a Costas sequence. A Welch sequence is one obtained from the Welch construction: for a prime $p$, a primitive element $\alpha\in\mathbb{F}_p$, and $c\in\mathbb{Z}$, the sequence $\alpha^{c},\alpha^{1+c},\dots,\alpha^{p-2+c}$. Golomb–Taylor's conjecture. A Costas sequence is singly periodical if and only if it is Welch. Only Welch sequences are currently known to be singly periodical, so the conjecture remains open.