Connelly's deformation-retraction conjecture for geodesic triangulations

Let $S$ be a closed orientable surface with a constant-curvature metric. Consider the space of geodesic triangulations of $S$ and the subgroup of isometries of $S$ homotopic to the identity. Connelly's conjecture. The space of geodesic triangulations of $S$ deformation retracts to the group of isometries of $S$ homotopic to the identity. This would describe the homotopy type of the deformation space of geodesic triangulations as the discrete analogue of the diffeomorphism group homotopic to the identity. The paper proves contractibility for negatively curved closed orientable surfaces, while the stated deformation-retraction claim is not resolved here.