Polyhedrality conjecture for graph functionals of compact Lie groups

Let $G$ be a sscc Lie group and let $S$ be a connected graph with $k$ vertices and $r$ edges. Write $d=\operatorname{rank}(G)$ and $l=\frac{\dim(G)-\operatorname{rank}(G)}{2}$, and let $\mathbb{Z}\Omega$ be the weight lattice of $G$. For each edge $e\in E(S)$, let $\lambda_e$ be a dominant weight, and let $\mathrm{GF}_S((\lambda_e)_{e\in E(S)})$ denote the associated graph functional. Graph-functional polyhedrality conjecture. There exists a sublattice $A_S\subset(\mathbb{Z}\Omega)^r$ of maximal rank $rd$ such that, if $(\lambda_e)_{e\in E(S)}\notin A_S$, then

$$\mathrm{GF}_S((\lambda_e)_{e\in E(S)})=0.$$

If $(\lambda_e)_{e\in E(S)}\in A_S$, then $\mathrm{GF}_S((\lambda_e)_{e\in E(S)})$ equals the number of integer points in a polytope $\mathscr{P}((\lambda_e)_{e\in E(S)})$ whose generic dimension is

$$\dim\mathscr{P}((\lambda_e)_{e\in E(S)})=lr-(2l+d)(k-1).$$

The dimension has this value whenever all the dominant weights $\lambda_e$ lie in the interior of the Weyl chamber; the defining equations of the polytope are affine in the weights, and the polytope is contained in the string cone $\mathscr{S}\!\mathscr{C}(G^r)$. This conjecture proposes a uniform polyhedral description of graph functionals, generalising the corresponding descriptions of dimensions and Littlewood--Richardson coefficients. The source presents it as a conjectural extension motivated by the circuit expansion for random geometric graphs; no resolution is supplied.