For every dimension d at least 4, a flat-manifold odometer yields a principal groupoid counterexample to the HK-conjecture, and no such d <= 3 example exists.
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A Low-Dimensional Counterexample to the HK-Conjecture
For every dimension d at least 4, a flat-manifold odometer yields a principal groupoid counterexample to the HK-conjecture, and no such d <= 3 example exists.