Goodrick's conjecture on collapsibility of star-shaped complexes

Let $C$ be a simplicial complex, and let $|C|$ denote its underlying space. The space $|C|$ is star-shaped if it is star-shaped in the relevant Euclidean space. Goodrick's conjecture. If $|C|$ is star-shaped, then $C$ is collapsible. This is one of several formulations of the longstanding open problem whether convex or star-shaped balls are collapsible; the source notes that the question has seen little progress beyond the three-dimensional case.