Generalized permutahedron property of the second Symanzik polynomial for exceptional Euclidean kinematics

Let $\mathcal{F}$ be the second Symanzik polynomial of a graph, and let $\mathbf{N}[\mathcal{F}]$ denote its Newton polytope. In the Euclidean regime, kinematics may be generic or exceptional.

Generalized permutahedron conjecture. The Newton polytope $\mathbf{N}[\mathcal{F}]$ is a generalized permutahedron in the Euclidean regime for all kinematics, including exceptional kinematics. Equivalently, Theorem~ holds without the generic-kinematics assumption.

For generic Euclidean kinematics this property is established, while the extension to exceptional kinematics is described as plausible but technically challenging and is left unproved.