Conjecture on the independence of the maps Pi_m and pi_m

For each integer $m\geq 0$, let $\Pi_m$ and $\pi_m$ be the maps defined in the paper from the Ekedahl–Oort stratification data. Independence conjecture. The maps $\Pi_m$ and $\pi_m$ are independent of $m\geq 0$.

The preceding results show inclusions and length bounds relating these maps as $m$ varies, including the case $m=0$. The conjecture asserts that this variation eventually disappears completely, but the supplied text gives no resolution.