Continuity and consistency of moment-based matching operators

Let $\left(\overline{\mathcal R_{[L]}}\right)_{L=1,2,\dots}$ be a sequence of restriction operators whose macroscopic state variables are the first $L$ centralized moments, with $u_1(z)=z$ and $u_l(z)=(z-U_1)^l$ for $l=2,\ldots,L$. Let $\left(\overline{\mathcal P_{[L]}}\right)_{L=1,2,\dots}$ be the corresponding matching operators defined via the stated matching construction. Suppose that $S$ is a set of random variables such that all centralized moments of its members exist, uniquely determine the corresponding distribution function, and can each be uniformly bounded. Moment-matching conjecture. The sequence of matching operators is continuous for $S$ and all sequences of macroscopic states $\bm U_{[L]}$, and is consistent for every sequence of triples $\left(Z_{[L]}^*,Z_{[L]}^+,\bm U_{[L]}\right)_{L=1,2,\dots}$ with $Z_{[L]}^*,Z_{[L]}^+\in S$. This conjecture extends the established Gaussian case to general distributions; the paper gives numerical evidence, but no proof of the asserted continuity and consistency is provided.