Leader–Radcliffe's variance-matched concentration conjecture for Bernoulli random variables

For each $\beta otin\frac12\boxtimes$? Let $T_i(\alpha_i)$ denote the random variables defined earlier in the paper, and let $\mathcal{Q}$ denote the concentration function used in Theorem. Assume that all the random variables below are independent. For any $\alpha_i\in[0,1]$, let $\alpha$ be the unique value in $[0,1]$ satisfying

$$\operatorname{Var}(T_1(\alpha_1)+\cdots+T_n(\alpha_n))=\operatorname{Var}(T_1(\alpha)+\cdots+T_n(\alpha)).$$

Leader–Radcliffe's conjecture.

$$\mathcal{Q}(T_1(\alpha_1)+\cdots+T_n(\alpha_n))\leq\mathcal{Q}(T_1(\alpha)+\cdots+T_n(\alpha)).$$

This conjecture proposes that, without the restriction $\alpha_i\geq1/2$, the concentration function is maximized by replacing the parameters with a common value chosen to preserve the total variance. The preceding discussion notes that the analogous assertion is known in the restricted case $\alpha_i\geq1/2$, while the unrestricted case is presented as open.