The more-choice conjecture for two-thinning

Let $n$ bins receive balls in the two-thinning setting, where the overseer may iteratively reject up to $k$ suggested allocations for each ball. More-choice conjecture. The asymptotically optimal maximum load is

$$\Theta\left(\left(\frac{\log n}{\log\log n}\right)^{1/(k+1)}\right).$$

This conjecture predicts that allowing more rejections improves the optimal maximum load in the two-thinning model. The preceding discussion contrasts this with the power-of-$k$ choices, where increasing the number of choices reduces the asymptotic maximum load only by a factor of $\log(k)$; the conjecture's resolution is not given here.