Stochastic lower-bound conjecture for quadratic Wald statistics

Let $f$ be a nonzero quadratic form and let $\Sigma$ be a nonzero positive semidefinite matrix. Write $W_{f,\Sigma}$ for the corresponding Wald random variable. Quadratic lower-bound conjecture. The distribution of $W_{f,\Sigma}$ stochastically dominates

$$\frac{1}{4}\,\chi^2_1.$$

A proposition proves this lower bound when the relevant eigenvalues are all nonnegative, while the conjecture concerns quadratic forms with both positive and negative eigenvalues. The authors say that the broader claim is suggested by their theorem and simulation experiments, but do not prove it.