The conjectured absolute-error threshold for Gaussian trace estimation

Let $m$ be the sampling number, let $\lambda=\lVert A\rVert_2$ and $\phi=\lVert A\rVert_F$ for a symmetric matrix $A$, and let $\varepsilon_{\mathrm{abs}}$ be the threshold appearing in the absolute-error theorem. Absolute-error threshold conjecture.

$$\varepsilon_{\mathrm{abs}}\leq\frac{2\lambda}{m}+\sqrt{\frac{2}{m}\phi^2+\left(\frac{2\lambda}{m}\right)^2}.$$

This would provide an explicit threshold for the absolute-error tail comparison with the extremal Gamma distribution. The source does not provide a resolution.