The hierarchical-order conjecture for uniform polymatroid access structures

Let $J$ be the ground set of a uniform polymatroid, let $\Pi=(P_x)_{x\in J}$ be a partition of the participants, and let $\Delta\subseteq\mathcal{P}(J)\setminus\{\emptyset\}$ be a monotone increasing family compatible with the polymatroid. Write $\Gamma(\Pi,\mathcal{Z},\Delta)$ for the induced access structure, and suppose the increment sequences satisfy

$$\boldsymbol{g}=(g_i)_{i\in I_m},\qquad \boldsymbol{g}'=(g'_i)_{i\in I_m}.$$

The hierarchical-order conjecture. If

$$\operatorname{sgn}(g_{i-1}-g_i)=\operatorname{sgn}(g'_{i-1}-g'_i)$$

for all $i=1,\ldots,m$, and $\Delta$ is compatible with both uniform polymatroids $\mathcal{Z}$ and $\mathcal{Z}'$, then the hierarchical preorders on $\Pi$ determined by $\Gamma(\Pi,\mathcal{Z},\Delta)$ and $\Gamma(\Pi,\mathcal{Z}',\Delta)$ are equal. The observation is based on computer calculations and concerns how the hierarchy depends on the signatures of consecutive differences in the increment sequence rather than on the particular rank values; it remains an unproved conjecture.