An improvement of the Boppana-Holzman bound for Rademacher random variables

Let $v_1,v_2,...,v_n$ be real numbers whose squares add up to $1$. Consider the $2^n$ signed sums of the form $S=\sum_{i=1}^n \pm v_i.$ Holzman and Kleitman (1992) proved that at least $\frac38=0.375$ of these sums satisfy $|S|\leq 1.$ By using bounds for appropriate moments of $S,$ Boppana and Holzman (2017) were able to improve the bound to $\frac{13}{32}=0.40625$ and even a bit better to $\frac{13}{32}+9\times10^{-6}.$ By following their approach, but using a key result of Bentkus and Dzindzalieta (2015), we will drastically improve (by more than 5\%) the latter barrier $\frac{13}{32}$ to $\frac{1}{2}-\frac{\Phi(-2)}{4\Phi(-\sqrt{2})}\approx 0.42768.$