de Groot's absolute cone conjecture

Let $X$ be an $n$-dimensional absolute cone, meaning a compact finite-dimensional metric space that is a cone with respect to every point. de Groot's absolute cone conjecture. The space $X$ is homeomorphic to the $n$-cell. The conjecture is true for $n\leq 4$ and false for $n\geq 5$, by the result attributed in the source to Guilbault.