Diameter conjecture for pointihedra

Let $P$ be a convex $n$-gon with $n\geq 13$, and let P^{\text{\raisebox{1px}{\scalebox{0.7}{\star}}}} be the resulting polygon with a puncture placed in its interior. Write {\overline{\mathcal F}}(P^{\text{\raisebox{1px}{\scalebox{0.7}{\star}}}}) for the pointihedron flip-graph and ${\rm diam}$ for graph diameter. Diameter conjecture. For some placement of the puncture in the interior of $P$,

$${\rm diam}({\overline{\mathcal F}}(P^{\text{\raisebox{1px}{\scalebox{0.7}{$\star$}}}}))=2n-7.$$

Earlier results in the paper give bounds on the diameter of these flip-graphs; this conjecture proposes the exact value $2n-7$ for a suitable placement when $n\geq 13$.