Automaton Game Correspondence

Let ${\mathrm{TB}}$ be the total budget, and let a unitary game be a game in the class defined by the paper. Associate to it the corresponding 0-bidding automaton $\mathcal{A}$, whose entries record the automaton's outcomes for budget partitions. Automaton Game Correspondence. The entries of $\mathcal{A}$ are obtained as outcomes of the game for every heap size at least

$$O({\mathrm{TB}}^2).$$

This conjecture asserts that sufficiently large heaps realize all entries of the corresponding automaton, strengthening the convergence-bound discussion. The supplied text does not indicate whether it has been resolved.