In a finite simplicial complex $K$, if the link of a simplex $\sigma$ is contractible then the two complexes $K$ and $K\setminus \text{Star}(\sigma)$ share the same homotopy type.
I am wondering if the converse is also true. Namely, if the link $\sigma$ of a simplicial complex $K$ is not contractible, is it true that $K$ and $K\setminus \text{Star}(\sigma)$ have different homotopy types?
I once post a similar question about how removing the star of a simplex in a simplicial complex alters its topology : Removing the star without changing homology but I am stuck for this question as well as it requires different tools. Indeed, I do not know how to show that the homotopy type of two spaces differs beside exhibiting an invariant that is not maintained (I did not have any in this case from my current topological toolbox).
I would really appreciate any comment, idea or reference on this question.