Existence conjecture for Butson Hadamard matrices of order $2p+4$

A Butson Hadamard matrix $BH(n,q)$ is an $n\times n$ complex matrix whose entries are $q$th roots of unity and whose rows are pairwise orthogonal under the standard complex inner product. Existence conjecture. For any prime $p$, Butson Hadamard matrices $BH(2p+4,2p)$ exist. The conjecture concerns the existence of a family of Butson Hadamard matrices at the parameters $(n,q)=(2p+4,2p)$; characterizing all pairs $(n,q)$ for which such matrices exist remains a fundamental open problem.