Given a $n \times n$ real Cauchy like matrix $C$, i.e., for real vectors $r$, $s$, $x$, $y$

$$ C_{ij} = \frac{r_i s_j}{ x_i - y_j} $$

Can a Cauchy-like $C$ be orthogonal, i.e., $C C^T = I$ for $n > 2$?

There exists such an orthogonal $C$ for $n = 2$ , $x = [1,0.4]$, $y = [6.25,0.625]$, $r = [-1.8114, 1.4811]$, and $s = [2.3367, -0.1225]$ with

$$ C = \begin{bmatrix} 0.8062 & 0.5916 \\ -0.5916 & 0.8062 \end{bmatrix} $$