The involutive linear-fractional bigrid perfection conjecture

Let $\tau$ be a linear fractional transformation of order $2$, and let a bigrid be an array with entries $\gamma(\epsilon,i,j)$ wherever they are defined. Let a bigrid be perfect in the sense defined in the paper. Involutive linear-fractional bigrid conjecture. Any bigrid satisfying

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

whenever both sides are defined is perfect. The source presents this as another conjectured component, obtained by computing tangent spaces to random elliptic bigrids over small finite fields; no general proof or disproof is given.