Usually, I like working with determinants related to the Vandermonde matrix, i.e.
$$\det(x_j^{i-1})=\prod_{i<j}(x_j-x_i).$$
However, I run into some unusual matrix and its determinant. Define the $(2n)\times (2n)$ matrix $\mathbb{M}_{2n}$ with entries
$$\mathbb{M}_{2n}(i,j)=
\begin{cases}
(x_j-x_i)(x_i-x_{2n}) \qquad \text{if $i<j$}, \\
(x_j-x_i)(x_j-x_{2n}) \qquad \text{if $i\geq j$}.
\end{cases}$$
>**QUESTION.** Is this true?
$$\det\mathbb{M}_{2n}=(x_1-x_2)^2(x_2-x_3)^2\cdots(x_{2n-1}-x_{2n})^2(x_{2n}-x_1)^2.$$