The Laplacian measure of a function in $G$ is the weak limit of linear combinations of the Laplacian measures of your elementary functions, whose support are "tripods", that is, unions of three rays emanating from a vertex.
In particular, the support of the Laplacian measure of a function in $G$ must be a union of tripods. Take a maximum of three independent linear functions that is lower bounded, and take the max of that and a positive number. This function has a Laplacian whose support cannot be expressed as a union of tripods.