A simplicial complex $S$ is collapsible if there is a sequence of elementary collapses that bring $S$ down to a single point; I'll denote this as $S \searrow \{pt\}$. I am wondering about a similar problem (the dual problem, in some ways).

If I have a simplicial complex $S \subset \Delta^n$, where $\Delta^n$ is the $n$-simplex, and there are elementary collapses bringing $\Delta^n$ to $S$, i.e. $\Delta^n \searrow S$, then $S$ must be contractible. Is the converse true: if $S$ is contractible, then is it true that $\Delta^n \searrow S$? How about if $S$ is collapsible?