Sturmfels–Thomas bounded-facet conjecture for Gröbner fans

Let $A\in{\bf Z}^{d\times n}$ be a matrix of rank $d$. The Gröbner fan of $A$ is the fan whose cones encode the reduced Gröbner bases of the toric ideal associated to $A$. Sturmfels–Thomas's conjecture. There exists a function $\varphi$ such that every cone of the Gröbner fan of $A$ has at most $\varphi(n-d)$ facets. This conjecture concerns uniform bounds on the combinatorial complexity of Gröbner fans in integer programming.