  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 $n \geq 2$ and
countable dense homogeneous. Furthermore, countable dense homogeneity can be
proven without assuming the space is connected.
  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 $X$ is strongly
$n$-homogeneous for all $n \geq 2$ and countable dense homogeneous.
