The finite-group criterion for holonomicity of quarter-plane walk generating functions

Let ${\mathcal Y}$ be a step set for walks in the quarter plane, and let $Q_{\mathcal Y}$ denote its generating function. Assume that the step set group ${\mathcal Y}$ is not singular. Holonomicity conjecture. The generating function $Q_{\mathcal Y}$ is holonomic if and only if the group of its step set is finite. This is known when $|{\mathcal Y}|=3$ and in all other cases known to the authors, but the general assertion remains open.