Descartes' rule of signs for arbitrary fewnomial systems

We consider systems of $n$ real polynomial equations in $n$ variables with $n+k+1$ monomials. By the Gale duality of Bihan and Sottile, their positive solutions correspond to the solutions of a system of $k$ equations $\prod_ip_i^{B_{ij}}=1$ in a polyhedron $Δ\subset\mathbb{R}^k$, where the $p_i$ ar