The maximum-weight cell-zeta conjecture for two-dimensional periods

Let $\Gamma$ be a two-dimensional completed primitive graph with $V$ vertices, and let $P^{(0)}_{\mathrm{max}}(\Gamma)$ denote the maximum-weight component of its period. Let $P_-(\Gamma)$ and $P_=(\Gamma)$ be respectively the holomorphic and antiholomorphic cell zeta values, with either value defined to be zero when the corresponding Hamiltonian cycle does not exist. Let ${\mathcal H}_{>0}^2$ denote the space of products of positive-weight multiple zeta values.

Maximum-weight cell-zeta conjecture.

$$P^{(0)}_{\mathrm{max}}(\Gamma)\equiv P_-(\Gamma)+P_=(\Gamma)\mod {\mathcal H}_{>0}^2.$$

The conjecture identifies the highest-weight part of a two-dimensional completed primitive period modulo products. It is motivated in the source by iterating the preceding lemma and is the precise cell-zeta formulation underlying the earlier forward reference.