A counterexample to conjecture "Catch 22" with 3 players, and 5 outcomes: 2 terminal and 3 cyclic

We construct a finite deterministic graphical (DG) game without Nash equilibria in pure stationary strategies. This game has 3 players I=\1,2,3\ and 5 outcomes: 2 terminal a₁ and a₂ and 3 cyclic. Furthermore, for 2 players a terminal outcome is the best: a₁ for player 3 and a₂ for player 1. Hence, the rank vector r is at most $(1,2,1)$. Here rᵢ is the number of terminal outcomes that are worse than some cyclic outcome for the player i ∈ I. This is a counterexample to conjecture ``Catch 22" from the paper ``On Nash-solvability of finite n-person DG games, Catch 22" (2021) arXiv:2111.06278, according to which, at least 2 entries of r are at least 2 for any NE-free game. However, Catch 22 remains still open for the games with a unique cyclic outcome, not to mention a weaker (and more important) conjecture claiming that an n-person finite DG game has a Nash equilibrium (in pure stationary strategies) when r = (0ⁿ), that is, all n entries of r are 0; in other words, when the following condition holds: • (C₀) any terminal outcome is better than every cyclic one for each player. A game is play-once if each player controls a unique position. It is known that any play-once game satisfying (C₀) has a Nash equilibrium. We give a new and very short proof of this statement. Yet, not only conjunction but already disjunction of the above two conditions may be sufficient for Nash-solvability. This is still open.