Unimodality conjecture for minimal filling neural-network architectures

Fix the depth $L$, the endpoint widths $d_0$ and $d_L$, and the activation degree $r$. Let $d53d=(d_0,d_1,d55fd53\dots,d_L)$ be a width vector, and let $d53d\preccurlyeqd53d'$ mean coordinatewise comparison. An architecture is filling when its neurovariety $d4b1_{d53d,r}$ has the full expected ambient dimension; $d53d$ is minimal when no smaller width vector with the same fixed endpoints has this property.

Unimodality conjecture. Every minimal width vector whose architecture is filling is unimodal: there exists $i\in\{0,1,d55fd53\dots,L\}$ such that $(d_0,d55fd53\dots,d_i)$ is weakly increasing and $(d_i,d55fd53\dots,d_L)$ is weakly decreasing.

The conjecture formalizes the architectural heuristic that a network should first expand its feature capacity and then contract it. The supplied text gives motivation but no proof or resolution.