PL equivalence of the 17-vertex and 16-vertex K3 triangulations

Let $(K3)_{16}$ and $(K3)_{17}$ denote the simplicial complexes with $16$ and $17$ vertices, respectively, constructed in the dilatation process described in the paper. The PL-equivalence conjecture. $(K3)_{17}$ is PL homeomorphic to $(K3)_{16}$. The authors were not able to prove this equivalence, but suggest it based on the fact that the given complex has only $17$ vertices; further experiments may establish the claim.