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

Let $v_1,v_2,\mathrm{.}\mathrm{.}\mathrm{.},v_n$ be real numbers whose squares add up to $1$. Consider the $2^n$ signed sums of the form $S={\sum }_{i=1}^n\pm v_i\mathrm{.}$ Holzman and Kleitman (1992) proved that at least $\frac{3}{8}=0.375$ of these sums satisfy $\mathrm{\mid }S\mathrm{\mid }\le 1.$ 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 $\frac{13}{32}+9\times {10}^{-6}\mathrm{.}$ 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 $\frac{1}{2}-\frac{\mathrm{\Phi }(-2)}{4\mathrm{\Phi }(-\sqrt{2})}\approx \mathrm{0.42768.}$