Given a smooth manifold $M$, There is known Kolar to be an equivalence between a linear, r-th order connection the tangent bundle $TM$ of $M$, and a principal connection on the r-th order frame bundle $P^r M$ over $M$.

I am interested in higher order frame bundles of the nonholonomic variety. That is to say if we consider the higher order frame bundle $P^r M$ to be bundle of invertible r-jets over $M$ at the origin $Inv(J^r M)$, then we would consider the nonholonomic frame bundle to be the bundle of invertible nonholonomic r-jets $Inv(\tilde{J}^r M)$ at the origin. Where nonholonomic jets indicate the iterative application of the 1-jet prolongation, i.e. $$\tilde{J}^r M = (J^1)^r M$$ Is anyone aware of any kind of analogous result to Kolars' for this case? I would expect an analogue as $\tilde{J}^r M$ is itself a principal bundle over $J^r M$ (see for example here.