The conjectured relative-error threshold for Gaussian trace estimation

Let $m$ be the sampling number, let $mu=r_{\mathrm{eff}}(A)$ for a nonzero symmetric positive semidefinite matrix $A$, and let $\varepsilon_{\mathrm{rel}}$ be the threshold appearing in the relative-error theorem. Relative-error threshold conjecture.

$$\varepsilon_{\mathrm{rel}}\leq\frac{2}{m\mu}.$$

This would give an explicit threshold beyond which the relative-error tail bound is controlled by the corresponding Gamma random variable. The source compares it with a bound valid for all errors but does not establish the conjectured threshold.