Jedwab–Pender conjecture on even-length quaternary Legendre pairs

A quaternary Legendre pair is a pair of quaternary sequences satisfying the Legendre-pair correlation conditions; here the length is an even integer $2N$.

Jedwab–Pender conjecture. There exists a quaternary Legendre pair of even length $2N$ for every $N\geq 1$.

Quaternary Legendre pairs generalize binary Legendre pairs and can be used to construct quaternary Hadamard matrices. Before this paper, the smallest open case of the conjecture was length $36$; the paper gives general constructions of even lengths but the universal assertion remains open in the supplied text.