Chen–Lev conjecture on Hilbert-cube pairs

Let $R_A(n)$ denote the number of representations of $n$ as a sum of two distinct elements of $A$. Let $H(d_{i_1},d_{i_2},\dots)$ be a half non-degenerated Hilbert cube, with $H_0(d_{i_1},d_{i_2},\dots)$ and $H_1(d_{i_1},d_{i_2},\dots)$ its even- and odd-part sets. Let $C$ and $D$ be different infinite sets of nonnegative integers with $0\in C$.

Chen–Lev conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exist positive integers $d_{i_1}<d_{i_2}<\cdots$ belonging to $D$ such that

$$C=H_0(d_{i_1},d_{i_2},\dots),$$ $$D=H_1(d_{i_1},d_{i_2},\dots).$$

The claim asks whether all such infinite pairs are generated by the two parity parts of a half non-degenerated Hilbert cube; the paper notes that the corresponding finite cases with parameters $l=0,1,2$ were established, while cases $l\ge3$ remained unsettled.