Kuperberg's mixed-base moment-matching conjecture

Let $p_1,p_2,\ldots,p_n \ge 2$ be a sequence of integers. Let $X_1,X_2,\ldots,X_n$ be independent random variables such that $X_k$ is uniformly distributed on the set

$$\\{p_k-1,p_k-3,p_k-5,\ldots,1-p_k\\}.$$

Kuperberg's mixed-base conjecture. There are unique constants

$$a_1>a_2>\cdots>a_n>0$$

such that, for

$$\widetilde{X}=a_1X_1+a_2X_2+\cdots+a_nX_n,$$

the first $2n$ moments of $\widetilde{X}$ agree with the first $2n$ moments of a random variable $Y$ uniformly distributed on $[-1,1]$, and

$$\sum_{j=1}^n (p_j-1)\left|a_j-\prod_{k=1}^j p_k^{-1}\right|<\prod_{k=1}^n p_k^{-1}.$$

This extends the paper's theorem from a common base to a mixed sequence of bases, and the asserted uniqueness and error bound give a precise form of the expected generalization. The supplied text does not report a proof or a resolution.