Possible orders of weighted two-dimensional combinatorial groups

Determine whether a two-dimensional walk model with non-trivial step weights can have a combinatorial group of order greater than 10, or prove that order 10 is the maximal possible order.

Background

The combinatorial group associated with a two-dimensional walk model is dihedral and may be finite or infinite. For unweighted models, the paper recalls that the finite possibilities have orders 4, 6, or 8; allowing non-trivial weights yields an example of order 10.

The authors state that no higher order is believed to occur, leaving unresolved whether finite weighted two-dimensional models with larger combinatorial groups exist. Establishing the maximal possible order would complete the finite-order classification in this weighted setting.