The conjectured dominating pair for DP-SGD with shuffled batches is not dominating

Chua, Ghazi, Kamath, Kumar, Manurangsi, Sinha and Zhang conjectured that the pair of Gaussian mixtures $P_{\mathcal S}=\frac1T\sum_t\mathcal{N}(2e_t,\sigma^2I)$, $Q_{\mathcal S}=\frac1T\sum_t\mathcal{N}(e_t,\sigma^2I)$ tightly dominates the adaptive batch linear query mechanism with shuffled batches, for every adaptive query method. We show that the conjecture fails for every number of steps $T\ge2$ and every batch size. The query method that breaks it answers as the conjectured worst case at every step but the last, and at the last step changes the query when the earlier outputs suggest that the differing record was in an early batch. At $T=2$, $\sigma=1$ and $\varepsilon=\frac12$ its hockey-stick divergence is $0.22323$ against $0.22127$ for the conjectured pair, and an interval-arithmetic certificate bounds the excess below by $1.04\cdot10^{-4}$. A direct analytic last-step argument gives strict failure at every $T\ge2$; a separate certificate at $T=3$ gives a quantitative bound. The same query method also beats the pair in Kullback–Leibler divergence and in Rényi divergence of every order we tested, from $\frac14$ to $5$, in both directions.