Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains after the open star of $\sigma$ is removed (that is, we remove $\sigma$ along with interiors of all cells $\tau$, if any, containing the closure of $\sigma$ in their boundary).

I'd like to be able to make the following claim as a small part of a proof I'm writing:

$S'$ is contractible.

Is this true? It certainly seems reasonable, and I could see a straightforward proof in the case where $S$ is a finite triangulation. If $S$ has only finitely many cells to begin with, I could obtain a triangulation by barycentric subdivision and then the desired statement holds. I'd really like to avoid that, but I worry about pathological examples like the Alexander horned sphere, etc.

So if I can't make the claim above, are there reasonable hypotheses to impose on $S$ (like: only finitely many cells allowed) which will make my desired claim true?