Circle-intersection Devron conjecture for polygons

Let $A$ be a $2n$-gon, with $n\geq 3$, such that circle intersection takes half of its vertices to a common point. Starting with the other half of the vertices, perform bipartite circle intersection.

Circle-intersection Devron conjecture. After $2n-6$ steps, one reaches another polygon for which circle intersection takes half of the vertices to a common point.

This is one of the experimentally observed settings in which the Devron property appears to persist despite the more complicated propagation rule. The conjecture includes the degenerate case $n=3$, where the conclusion follows from a Euclidean-geometry result about three circles having a common point.