Kiss–Rozgonyi–Sándor conjecture on higher-order representation functions

Let $h>2$, and let $C$ and $D$ be different infinite sets of nonnegative integers. For a set $A$, write $A(z)=\sum_{a\in A}z^a$, and let $F_C(z)$, $F_D(z)$, and $T(z)$ denote the generating polynomials of the finite sets $F_C$, $F_D$, and $T$, respectively. The functions $R_{h,C}^{(1)}$ and $R_{h,D}^{(1)}$ count ordered representations as sums of $h$ elements of the indicated set.

Kiss–Rozgonyi–Sándor conjecture. The equality $R_{h,C}^{(1)}(n)=R_{h,D}^{(1)}(n)$ from some point onward holds if and only if there exist positive integers $n_0$ and $M$, and finite sets $F_C,F_D,T$ satisfying $F_C\cup F_D\subset[0,Mn_0-1]$ and $T\subset[0,M-1]$, such that

$$C=F_C\cup\{lM+t:l\ge n_0,\,t\in T\},$$ $$D=F_D\cup\{lM+t:l\ge n_0,\,t\in T\},$$ $$(1-z^M)^{h-1}\mid(F_C(z)-F_D(z))T(z)^{h-1}.$$

This conjectures a periodic-tail characterization of pairs of infinite sets whose ordered $h$-fold representation functions eventually coincide, generalizing Nathanson's result for $h=2$.