The claimed classification of orders for Hadamard matrices with one circulant core

A Hadamard matrix with one circulant core of order $p+1$ has the form

$$H=\begin{pmatrix}1&e\\ e^{t}&A_{C}\end{pmatrix},$$

where $e$ is the all-one row vector of dimension $p$, $e^{t}$ is its transpose, and $A_{C}$ is a circulant matrix of order $p$. The source lists the following sufficient constructions for a Hadamard matrix with circulant core: $p\equiv3\pmod 4$ prime; $p=q(q+2)$ with $q$ and $q+2$ prime; $p=2^{t}-1$ for positive integer $t$; or $p=4x^{2}+27$ with $p$ prime and $x$ a positive integer.

One-circulant-core order claim. These are the only possible orders for a Hadamard matrix with one circulant core.

The source states this as a conjectural classification, but supplies no proof or resolution. Its precise scope is ambiguous because the listed parameter $p$ is the core-related order while the matrix itself has order $p+1$.