Vertex-minimal triangulation of the Poincaré homology 3-sphere

Prove that every simplicial triangulation of the Poincaré homology 3-sphere requires at least 16 vertices; equivalently, establish that the 16-vertex triangulation constructed by Björner and Lutz is vertex-minimal among all triangulations of the Poincaré homology 3-sphere.

Background

In this paper the authors construct convex cocompact right-angled reflection groups acting on H^n whose limit sets realize various three-dimensional continua. For item (3) of Corollary 1.3, they rely on the 16-vertex triangulation of the Poincaré homology 3-sphere due to Björner and Lutz to obtain a convex cocompact reflection group in Isom(H^16) with limit set a Čech cohomology 3-sphere not homeomorphic to S^3.

They note that Björner and Lutz conjectured that 16 is the minimal number of vertices required to triangulate the Poincaré homology 3-sphere, and record the best known general lower bound (at least 12 vertices for any homology 3-sphere distinct from S^3). A resolution of this conjecture would settle the vertex-minimality of the known triangulation and directly impacts the minimal ambient hyperbolic dimension achievable by their construction, since the dimension equals the number of vertices in the chosen triangulation.