Bounded-facet conjecture for cones of Gale-transform fans

Let $A\in{\bf Z}^{d\times n}$ be a unimodular matrix of rank $d$, and let $B$ be a Gale transform of $A$. The fan $\Gamma(B)$ is the fan associated with the Gale transform $B$. Specialized bounded-facet conjecture. There exists a function $\varphi'$ such that every cone of $\Gamma(B)$ has at most $\varphi'(n-d)$ facets. This is the unimodular specialization of the bounded-complexity conjecture for Gröbner fans, using the identification of the Gröbner fan of $A$ with $\Gamma(B)$ in the unimodular case.