A finite three-ample complex with a nontrivial fundamental group

We construct a finite three-ample simplicial complex on $32{,}768$ vertices which retracts simplicially onto a four-cycle. Thus the prescribed extension of every subcomplex on at most three vertices is compatible with a nontrivial fundamental group in a finite complex. The construction fixes a four-cycle and chooses the remaining permitted faces probabilistically. A finite union bound proves that all extension requests can be met at once.