General concentration conjecture for bounded-atom random vectors

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{R}^d$ satisfying

$$\sup_{x\in \mathbb{R}^d}\mathbb{P}(X_i=x)\leq \alpha.$$

A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and all $x\in \mathbb{R}^d$,

$$\mathbb{P}(X_1+\ldots+X_n = x)\leq \max_{k\in \mathbb{Z}}\mathbb{P}(w_1U^{\alpha}_1+\ldots+w_nU^{\alpha}_n=k).$$

This is stated as a second conjecture concerning the paper's theorem; the notation $U^{\alpha}_i$ is not defined in the supplied context, and no resolution is reported.