Inverse-chi-square quadratic-form conjecture for weighted normal reciprocals

Let $X=(X_1,X_2,\ldots,X_k)^T\sim\mathcal{N}_k(0,\Sigma)$, with none of its entries a point mass. Let $p_1,p_2,\ldots,p_n$ be nonnegative numbers satisfying

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

Inverse-chi-square conjecture. Then

$$\left(\frac{p_1}{X_1},\frac{p_2}{X_2},\ldots,\frac{p_n}{X_n}\right)\Sigma \left(\frac{p_1}{X_1},\frac{p_2}{X_2},\ldots,\frac{p_n}{X_n}\right)^T \sim\frac{1}{\chi^2_1}.$$

The paper presents this as an equivalent reciprocal formulation of the weighted-ratio Cauchy conjecture. Its notation uses $k$ for the dimension of $X$ but $n$ for the number of weights, so that indexing should be checked against the source.