Duchêne–Rigo conjecture on invariant games for complementary Beatty sequences

Let $S=(A_n,B_n)_{n\geq 1}$ be a pair of complementary Beatty sequences, and let an invariant game mean a combinatorial game whose legal moves are invariant under translation by game positions. A position is a $P$-position if the previous player has a winning strategy.

Duchêne–Rigo conjecture. There exists an invariant game having

$$S\cup\{(0,0)\}$$

as its set of $P$-positions.

This conjecture addresses the inverse problem of constructing games from prescribed complementary sequences. It was proven by Larsson, Hegarty, and Fraenkel in 2011.