Uniqueness of section-free equations for finite-orbit models

Let $d=2$ and consider a model whose orbit is finite. A section-free equation is a linear combination of the orbit equations in which the boundary sections do not occur. If the model has no large forward step, then the uniqueness conjecture. there exists a unique section-free equation, up to multiplication by a nonzero constant. This concerns the extension of the one-dimensional uniqueness result to dimension two; existence is asserted for finite orbits, while uniqueness in the stated two-dimensional setting remains open.