The open-ball conjecture for full arc complexes of decorated once-punctured polygons

Let $n\geq2$, and let $\depu n$ be a decorated once punctured $n$-gon. Its full arc complex is the simplicial complex whose simplices consist of collections of pairwise compatible arcs in $\depu n$. The open-ball conjecture. The full arc complex of $\depu n$ is homeomorphic to an open ball of dimension $2n-4$. This conjecture describes the global topology of the arc complex and is part of the paper's proposed picture for decorated punctured polygon arc complexes; the supplied text does not give a resolution.