The one-circulant-core Hadamard matrix conjecture

Let $p$ be a positive integer. A Hadamard matrix with one circulant core of order $p+1$ has block form $\left(\begin{smallmatrix}1&e\\e^{t}&X_C\end{smallmatrix}\right)$, where $e=(1,1,\ldots,1)$ has dimension $p$ and $X_C$ is a circulant matrix of order $p$. Such a matrix can be constructed when one of the listed sufficient conditions holds: $p\equiv3\pmod 4$ is prime; $p=q(q+2)$ with $q,q+2$ prime; $p=2^t-1$; or $p=4x^2+27$ with $p$ prime. The one-circulant-core Hadamard matrix conjecture. If a Hadamard matrix with one circulant core of order $p+1$ exists, then $p$ satisfies one of those four conditions. The claim proposes that the known sufficient constructions exhaust the possible parameters, but the source gives no resolution.