  The acyclic system conjecture of Ardila and Ceballos can be interpreted as
saying the following: "Every triangulation of the 3-skeleton of a product of
two simplices can be extended to a triangulation of the whole product". We show
a counter-example to this.
  Motivation for this conjecture comes from a related conjecture, the
"spread-out simplices" conjecture of Ardila and Billey. We give some necessary
conditions that counter-examples to this second conjecture (if they exist) must
satisfy.
