Chromatic recurrence in intersections with H-sets

Let $S\subseteq\mathbb Z$ be a set of chromatic recurrence, let $k\in\mathbb N$, let $C>0$, and let $\widetilde H(\boldsymbol\alpha;C\sqrt d,\varepsilon)$ denote the set defined earlier in the paper. Intersection recurrence conjecture. There exist infinitely many $d\in\mathbb N$ and $\boldsymbol\alpha\in\mathbb T^d$ such that, for every $\varepsilon>0$,

$$S\cap\widetilde H(\boldsymbol\alpha;C\sqrt d,\varepsilon)$$

is a set of $k$-chromatic recurrence. The conjecture is proposed as the key generalization needed for a possible extension of the proof of the paper's main theorem; no resolution is given.