Before revision:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
After revision:  This matches the situation for the subfamily of regular triangulations, ie. partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.