We say that a simplicial complex $K$ is acyclic if it's integral reduced simplicial homology groups are trivial in all dimensions.

For a vertex ${v} \in K$, we define the link

$$lk(v) :=\{\sigma \in K \; | \; \sigma \cup \{v\} \in K, \sigma \cap \{v\} = \emptyset\}.$$

A simplicial complex is an integral generalised homology $n$-sphere if it has the homology of $S^n$, and has the same integral local homology groups as $\mathbb{R}^n$.

There exist acyclic finite simplicial complexes such that the link of every vertex is non-acyclic. Does there exist a finite acyclic simplicial complex for which the link of every vertex is itself an integral generalised homology sphere? Alternatively, such that the link of every vertex has the homology of a sphere?