The adjacent-row bigrid perfection conjecture

Let a bigrid be an array with entries $\gamma(\epsilon,i,j)$, where the indices are taken wherever the entries are defined, and let a bigrid be perfect in the sense defined in the paper. Adjacent-row bigrid perfection conjecture. Any bigrid satisfying

$$\gamma(1,i,j)=\gamma(0,i+1,j)$$

whenever both sides are defined is perfect. The source notes that this conjecture is straightforward to verify when $m=n=2$, but gives no general resolution.