Infinite Separation between General and Chromatic Memory

In this paper, we construct a winning condition $W$ over a finite set of colors such that, first, every finite arena has a strategy with 2 states of general memory which is optimal w.r.t.~$W$, and second, there exists no $k$ such that every finite arena has a strategy with $k$ states of chromatic memory which is optimal w.r.t.~$W$.