On homogeneous locally conical spaces
The main result of this article is:
Theorem. Every homogeneous locally conical connected separable metric space that is not a $1$-manifold is strongly $n$-homogeneous for each $n \geq 2$. Furthermore, every homogeneous locally conical separable metric space is countable dense homogeneous.
This theorem has the following two consequences.
Corollary 1. If $X$ is a homogeneous compact suspension, then $X$ is an absolute suspension $($i.e., for any two distinct points $p$ and $q$ of $X$ there is a homeomorphism from $X$ to a suspension that maps $p$ and $q$ to the suspension points$)$.
Corollary 2. If there exists a locally conical counterexample $X$ to the Bing–Borsuk Conjecture $($i.e., $X$ is a locally conical homogeneous Euclidean neighborhood retract that is not a manifold\/$)$, then each component of $X$ is strongly $n$-homogeneous for all $n \geq 2$ and $X$ is countable dense homogeneous.