Hereditary chromatic recurrence inside density-recurrent sets

Let $S\subseteq\mathbb Z$ be a set of density recurrence, meaning that for every subset $A$ of a suitable positive upper density, $(A-A)\cap S\neq\varnothing$. Hereditary recurrence conjecture. There exists a set $S'\subseteq S$ such that $S'$ is a set of chromatic recurrence and is not a set of density recurrence. This conjecture concerns whether every density-recurrent set contains a chromatically recurrent subset that loses density recurrence; the source explicitly presents it as an open question related to the paper's results.