Lattice-valued random-vector concentration conjecture

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{Z}^d$. A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and all $x\in \mathbb{R}^d$,

$$\mathbb{P}(a_1X_1+\ldots+a_nX_n = x)\leq \max_{k\in \mathbb{Z}^d}\mathbb{P}(w_1X_1+\ldots+w_nX_n=k).$$

The paper presents this as a more general result for lattice-valued random vectors; by Corollary 1 it would suffice to prove it for odd $n$, and its resolution is not given here.