Chen–Lev strengthened conjecture on explicit Hilbert-cube pairs

Let $C$ and $D$ be different sets of nonnegative integers satisfying

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

for integers $r\ge0$ and $m\ge2$, and let $R_C(n)$ and $R_D(n)$ denote their two-element representation functions.

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

$$C=H_0(1,2,4,8,\dots,2^{2l-1},2^{2l}-1,2^{2l+1}-1,2(2^{2l+1}-1),4(2^{2l+1}-1),8(2^{2l+1}-1),\dots),$$ $$D=H_1(1,2,4,8,\dots,2^{2l-1},2^{2l}-1,2^{2l+1}-1,2(2^{2l+1}-1),4(2^{2l+1}-1),8(2^{2l+1}-1),\dots).$$

This is explicitly described as a stronger version of the Chen–Lev question and would identify the sets themselves, not only the parameters of their periodic intersection. The source supplies no resolution.