Ratio of independent normals is Cauchy — $W$ is not chi-square

Let $(X,Y)\sim N(\underline0,\Sigma)$ where $\Sigma=\left(\begin{smallmatrix}1&\rho\\\rho&1\end{smallmatrix}\right),\ \rho\neq0$ and $W=\dfrac{X-Y}{X+Y}\sqrt{\dfrac{1+\rho}{1-\rho}}$. Then which of the following statements is not correct?