Katznelson's recurrence conjecture for iterated differences in $\mathbb Z^\omega$

Let $\mathbb Z^\omega$ be the direct sum of countably many copies of $\mathbb Z$, with standard basis $\{\mathbf e_j:j\in\mathbb N\}$. Write $\mathcal E_1=\{\mathbf e_j:j\in\mathbb N\}$, and let $\Delta_2(\mathcal E_1)$ denote the set of differences $(a-b)-(c-d)$ with $a,b,c,d\in\mathcal E_1$ mutually distinct. A set is a set of Bohr recurrence if it intersects every Bohr neighborhood of $0$, and a set of chromatic recurrence if its associated Cayley graph has infinite chromatic number.

Special Katznelson conjecture. If $S\subseteq\Delta_2(\mathcal E_1)$ is a set of Bohr recurrence in $\mathbb Z^\omega$, then $S$ is a set of chromatic recurrence.

This is a special case of the general recurrence problem and is used in the paper as a sufficient statement for reducing the question to arbitrary countably infinite abelian groups.