The generic total H-complexity conjecture for ReLU neural network maps

Let $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be a ReLU neural network map with canonical polyhedral decomposition $\mathcal{C}(F)$. The total $H$-complexity of $F$ and the $0$-cells of $\mathcal{C}(F)$ are understood as defined in the paper.

Generic total H-complexity conjecture. With probability $1$, the total $H$-complexity of $F$ is less than or equal to the number of $0$-cells of $\mathcal{C}(F)$.

The conjecture is motivated by the contrast between high-complexity PL examples with a single $0$-cell and the claim that such behavior is highly atypical for ReLU neural network functions. The supplied text gives no resolution.