Chen–Lev conjecture on periodic intersections of representation-equivalent sets

Let $C$ and $D$ be different sets of nonnegative integers, and let $R_C(n)$ count representations of $n$ as a sum of two distinct elements of $C$, with the analogous definition for $D$. Assume

$$C\cup D=\mathbb N,$$ $$C\cap D=r+m\mathbb N,$$

where $r\ge0$ and $m\ge2$ are integers.

Chen–Lev conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exists an integer $l\ge1$ such that

$$r=2^{2l}-1,\qquad m=2^{2l+1}-1.$$

This question seeks a classification of covering pairs with periodic intersection and identical two-element representation functions. The source presents the assertion as a question posed by Chen and Lev; no resolution is supplied.