@Andy: there is now a small explicit example available, with 15 vertices and 55 tetrahedra, in case you're still interested: it's the 3-ball called $B_{15,66}$ in [here][1]. (Sorry for the huge delay and the shameless self-promotion.)


  [1]: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i3p31