Algebraic representability by rational absorbing games

An absorbing game has a limit value, and a rational absorbing game is one whose payoff and transition data are rational. A number is algebraic if it is a root of a nonzero polynomial with rational coefficients. Rational algebraic representability conjecture. Any algebraic number can be represented as the limit value of a rational absorbing game. The paper establishes rational absorbing games with irrational limit values, including a $3\times3$ example of algebraic degree three and $2\times2$ examples realizing every positive square root; representing every algebraic number remains open.