The canonical-weight criterion for the $\tau_B=\tau_R$ condition

Let a Newton polynomial have maximally tempered coefficients, and let the same coefficients be obtained from canonical edge weights on the associated brane tiling. Let $\tau_B$ and $\tau_R$ denote the complex structures associated with the blue and red data, respectively. Equality of these complex structures is considered modulo the action of $\operatorname{SL}(2,\mathbb{Z})$.

Canonical-weight criterion conjecture. Up to $\operatorname{SL}(2,\mathbb{Z})$, the condition

$$\tau_B=\tau_R$$

holds if and only if the maximally tempered coefficients of the Newton polynomial coincide with the coefficients from canonical edge weights on the tiling.

The conjecture proposes that the symmetry reflected by equality of the two complex structures is exactly characterized by agreement between maximally tempered and canonical-weight coefficients. The claim is based on known examples, while examples such as the suspended pinch point motivate the distinction.