The polyhedral realization conjecture for the bialgebra PROP differential

Let $m,n$ be positive integers, let $\xi^m_n$ denote the generator indexed by $m$ inputs and $n$ outputs, and let $\partial$ be the perturbed differential in the minimal model of the PROP for bialgebras. A convex polyhedron of dimension $m+n-3$ has codimension-one faces of dimension $m+n-4$. Polyhedral realization conjecture. There exists a series of convex $(m+n-3)$-dimensional polyhedra $B^m_n$ such that

$$\partial(\xi^m_n)$$

is the sum of the codimension-one faces of these polyhedra. This conjecture proposes a geometric realization of the differential in terms of convex polyhedra, extending the explicit low-complexity calculations in the resolution of the PROP for bialgebras. The supplied text gives no evidence that the conjecture has been proved or refuted.