Cauchy distribution conjecture for weighted ratios of correlated normals

Let $X=(X_1,X_2,\ldots,X_k)^T$ and $Y=(Y_1,Y_2,\ldots,Y_k)^T$ be independent random vectors with the same distribution $\mathcal{N}_k(0,\Sigma)$, where $\Sigma$ has positive diagonal entries. Let $p_1,p_2,\ldots,p_k$ be nonnegative numbers satisfying

$$p_1+p_2+\cdots+p_k=1.$$

Cauchy ratio conjecture. The random variable

$$\frac{p_1Y_1}{X_1}+\frac{p_2Y_2}{X_2}+\cdots+\frac{p_kY_k}{X_k}$$

has the standard Cauchy distribution. This is presented as an equivalent formulation of the monomial conjecture. The paper notes that an induction argument involving ratios of normal variables with nonzero means was considered but not successfully carried out.