Geometric Newton-polytope formulation of the maximal-ReLU-function conjecture

Let $\Delta(m)$ be the recursively defined collection of network polytopes with $m$ hidden layers, and let $\mathcal{N}(f)$ denote the Newton polytope of a positively homogeneous CPWL function $f$. Set

$$m=\lceil\log_2(n+1)\rceil-1.$$

Hertrich et al.'s geometric conjecture. There do not exist polytopes $P,Q\in\Delta(m)$ such that

$$P=\mathcal{N}\bigl(\max\{x_1,x_2,\ldots,x_n,0\}\bigr)+Q.$$

This is described as an equivalent geometric formulation of the conjecture that the displayed maximal function requires depth $\lceil\log_2(n+1)\rceil$. Its general status is open.