On a multidimensional spherically invariant extension of the Rademacher--Gaussian comparison

It is shown that \begin{equation*} $\mathsf{P}$(|a_1U_1+\dots+a_nU_n|>u)\le c,$\mathsf{P}$(a|Z_d|>u) \end{equation*} for all real $u$, where $U_1,\dots ,U_n$ are independent random vectors uniformly distributed on the unit sphere in ${\mathbb{R}}^d$, $a_1,\dots ,a_n$ are any real numbers, $a:=\sqrt{(a_1^2+\cdots +a_n^2)\mathrm{/}d}$, $Z_d$ is a standard normal random vector in ${\mathbb{R}}^d$, and $c=2e^3\mathrm{/}9=4.46\dots$. This constant factor is about $89$ times as small as the one in a recent result by Nayar and Tkocz, who proved, by a different method, a corresponding conjecture by Oleszkiewicz. As an immediate application, a corresponding upper bound on the tail probabilities for the norm of the sum of arbitrary independent spherically invariant random vectors is given.