Toward the best constant factor for the Rademacher-Gaussian tail comparison

Let S_n:=a_1\vp_1+…+a_n\vp_n, where \vp_1,…,\vp_n are independent Rademacher random variables (r.v.'s) and a_1,…,a_n are any real numbers such that a_1^2+…+a_n^2=1. Let Z be a standard normal r.v. It is proved that the best constant factor c in inequality \P(S_n>x) \leq c\P(Z>x) for all x in \R is between two explicitly defined absolute constants c_1 and c_2 such that c_1<c_2 \approx 1.01c_1.