Absorbing-game limit values of maximal algebraic order

Let $m\geq 1$. A rational absorbing game with $m$ actions per player is an absorbing game whose data are rational and in which each player has $m$ available actions. Maximal-degree conjecture. For all $m\geq 1$, there exists a rational absorbing game with $m$ actions per player whose limit value is algebraic of order $m$. Here, “algebraic of order $m$” means algebraic of degree $m$ over the rationals. The result is motivated by the paper's examples: games with fewer than three actions on one side have rational limit value, while a $3\times3$ example has an irrational limit value of algebraic degree three; the asserted degree-$m$ construction for every $m$ remains open.