Brown–Graham–Landman conjecture on 2-large and large sets

A set $S\subset\mathbb{N}$ is large if it is a set of $d$-topological recurrence for every $d$, and it is 2-large if every 2-coloring of $\mathbb{N}$ has arbitrarily large arithmetic progressions whose common difference lies in $S$.

Brown–Graham–Landman conjecture. If $S$ is not a set of $d$-nilBohr recurrence for some $d$, then $S$ is not $2$-large.

Brown, Graham, and Landman asked whether every 2-large set is large. The paper has constructed a set that is not large but meets every nil-Bohr set, and conjectures that such nil-Bohr recurrence is necessary for a counterexample to the 2-large versus large question.