The more-balls conjecture for two-thinning

Let $m=\Omega(n)$ balls be allocated to $n$ bins in the two-thinning setting, and let the overseer two-thin the allocations. More-balls conjecture. The asymptotically optimal maximum load is

$$\frac{m}{n}+\Theta\left(\sqrt{\frac{\log n}{\log\log n}}\right).$$

This conjecture concerns the heavily loaded regime, where the number of balls is at least of order $n$, and predicts the optimal deviation of the maximum load above the average load $m/n$. The source compares it with results for power-of-$k$ choices in the heavily loaded balls-and-bins model; its resolution is not given here.