The two-circulant-core Hadamard matrix conjecture

Let $n$ be a positive integer. A Hadamard matrix of order $2n+2$ with two circulant cores is a matrix whose two core blocks $A$ and $B$ are circulant matrices of order $n$ and satisfy $AA^{t}+BB^{t}=(2n+2)I_n-2J_n$. The two-circulant-core Hadamard matrix conjecture. For every odd $n$, there exists a Hadamard matrix of order $2n+2$ with two circulant cores. Several infinite families and sporadic values are known, but the assertion for every odd $n$ is unresolved in the source.