The planar open-book conjecture for tight positive contact structures on Brieskorn homology spheres

Let $Y$ be a Brieskorn homology sphere equipped with a tight, positive contact structure $\xi$. A planar open book decomposition is an open book supporting $\xi$ whose pages have genus zero.

Planarity conjecture. No tight, positive contact structure on a Brieskorn homology sphere is supported by a planar open book decomposition.

The conjecture is motivated by the preceding theorem, which rules out planarity under two alternative hypotheses, and by the authors' observation that they know no counterexample outside those cases. The source does not establish the conjecture in full.