The oriented-matroid determination conjecture for polytopal measure cones

Let $S$ be a finite spanning set, and let $\mathfrak M_\mathbb Z(S)$ be the lattice generated by standard measures of simplices with vertices in $S$. Define the rational cone $\mathfrak{Pos}(S)\subset\mathfrak M_\mathbb Z(S)$ as the cone of non-negative linear combinations of the standard measures $\mu_\mathcal P$ of all generalized polytopes $\mathcal P$ whose vertices are contained in $S$. Let the oriented matroid associated to $S$ encode the orientations of the affine dependencies among points of $S$. Oriented-matroid determination conjecture. The rational cone $\mathfrak{Pos}(S)$ is uniquely determined by the oriented matroid associated to $S$. The supplied text poses this as an open claim about the combinatorial determination of the cone; no resolution is provided.