Bousquet-Mélou–Mishna D-finiteness conjecture for quarter-plane walks

A small-step quarter-plane lattice model is determined by a stepset $S\subseteq\{-1,0,1\}^2\setminus\{(0,0)\}$, and its counting generating function is

$$Q_S(t)=\sum_{\omega}t^{|\omega|},$$

where the sum ranges over walks starting at the origin, using steps in $S$, and remaining in $\mathbb{N}^2$. A model is called nontrivial if it is not one of the excluded degenerate cases, and models are considered up to isomorphism. Bousquet-Mélou–Mishna conjecture. For each of the 79 nontrivial non-isomorphic unweighted small-step quarter-plane models, $Q_S(t)$ satisfies a linear differential equation with polynomial coefficients—that is, it is D-finite in $t$—if and only if the particular group associated with the walk is finite. This conjecture concerns the classification of D-finite counting generating functions for the 79 models; the cited context presents it as the conjecture being examined, without resolving its status.