Monomial Wald statistic distribution conjecture

Let $\Sigma$ be any positive semidefinite $k\times k$ matrix with positive diagonal entries. For nonnegative real exponents $\alpha_1,\ldots,\alpha_k$ that are not all zero, define the monomial

$$f(x_1,\ldots,x_k)=x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_k^{\alpha_k}.$$

Monomial distribution conjecture. The Wald variable satisfies

$$W_{f,\Sigma}\sim\frac{1}{(\alpha_1+\cdots+\alpha_k)^2}\,\chi^2_1.$$

The conjecture asserts that this distribution is independent of the covariance matrix apart from the stated positive-diagonal condition. The paper explains that it is motivated by established results for bivariate monomials and discusses it as unresolved.