Algebraicity conjecture for Kreweras and reverse Kreweras walks

Let $G(t;a,b)$ denote the generating function for the corresponding model, with parameters $t$, $a$, and $b$. Models 19 and 20 are the Kreweras and reverse Kreweras walks, respectively. A generating function is algebraic if it satisfies a nonzero polynomial equation with coefficients in the relevant rational function field. Algebraicity conjecture. The generating functions $G(t;a,b)$ for Models 19 and 20, Kreweras and reverse Kreweras walks, are algebraic for all $a,b$.

The generating functions for Models 19 and 20 are equal, and algebraicity has been shown when $a=b$. Algebraic solutions have also been guessed at various integer values of $a$ and $b$, while the general case remains unresolved.