Finite verification conjecture for rational discrete equidecomposability of polygons

Let $P$ and $Q$ be rational polygons. The facet criterion in condition (3) of the main result, involving a countable family of discrete dynamical systems generated by minimal triangulations, is one of the three necessary and sufficient conditions for rational finite discrete equidecomposability.

Finite verification conjecture. Checking condition (3) can be reduced to a finite number of verifications. In particular, there exists a finite procedure for determining whether $P$ and $Q$ are rationally discretely equidecomposable.

The conjecture would make the criterion effectively decidable by replacing the countable family of dynamical-system checks with finitely many verifications. The paper presents the necessary and sufficient criteria but leaves this reduction to finite verification open.