**Definition 1**: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.

**Definition 2**: A matrix $A$ is half-skew-centrosymmetric if there exist two square matrices $B$ and $C$ of order $n$ such that 

  \begin{equation}
      A = \begin{bmatrix}
        B & R C R \\
        C & -R B R
      \end{bmatrix}.
  \end{equation}

where $R$ is the reverse identity matrix.

One day I found that these two definitions can be considered simultaneously. For example, \begin{bmatrix}
        1 & 1 \\
        1 & -1
      \end{bmatrix} is the simplest half-skew-centrosymmetric Hadamard matrix.

Here comes my questions:

>
> First, can you give some references about half-skew-centrosymmetric Hadamard matrices? Are there other names for these matrices?  

> Second one is about construction methods. It is easy to figure out how to construct a half-skew-centrosymmetric Hadamard matrix of order $2^k\cdot n$ based on a Hadamard matrix of order $n$ by using a variant of Sylvester's construction.
 Can you propose more methods to construct half-skew-centrosymmetric Hadamard matrices?

> Third, can you prove the following conjecture or give a counter-example?
>
> **Conjecture**: A half-skew-centrosymmetric Hadamard matrix exists for $n=2$ or $n$ is a multiple of $4$.