Positive-density reciprocal conjecture for uniformly distributed sets

Let $A\subseteq\mathbb{N}$ be a set containing $0$. Say that $A$ is uniformly distributed modulo every power of $2$ if, for every $k\geq1$, its elements are uniformly distributed among the residue classes modulo $2^k$. Say that $A$ is periodic if there exists a positive integer $m$ such that membership in $A$ depends only on the residue modulo $m$. Its reciprocal is

$$\overline{A}:=\{n\in\mathbb{N}:n-A\subseteq A\}.$$

Positive-density reciprocal conjecture. If $A$ contains $0$, is not periodic, and is uniformly distributed modulo every power of $2$, then $\overline{A}$ has positive density. The conjecture identifies a broad class of nonperiodic, 2-adically uniform sets whose reciprocals should be substantially larger than those of thin structured examples. The paper states it as the strongest conjecture consistent with its theorems and experiments; no resolution is given in the supplied text.