Geometric Hermite interpolation with maximal order and smoothness

We conjecture that, under suitable assumptions, splines of degree less than or equal to n can interpolate points on a smooth curve in R(m) with order of contact k-1=n-1+[(n-1)/(m-1)] at every nth knot. Moreover, this Geometric Hermite Interpolant (GHI) has the optimal approximation order k+1. We give a proof of this conjecture for planar quadratic spline curves and describe a simple construction of curvature continuous quadratic splines from control polygons.