Given a smooth manifold $M$, Kolář - On the torsion of linear higher order connections showed that there is 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 $\operatorname{Inv}(J^r M)$, then we would consider the nonholonomic frame bundle to be the bundle of invertible nonholonomic $r$-jets $\operatorname{Inv}(\smash{\tilde{J}}^r M)$ at the origin. Where nonholonomic jets indicate the iterative application of the 1-jet prolongation, i.e. $$\smash{\tilde{J}}^r M = (J^1)^r M$$ Is anyone aware of any kind of analogous result to Kolář's for this case? I would expect an analogue as $\smash{\tilde{J}}^r M$ is itself a principal bundle over $J^r M$ (see for example de Léon and Martin Méndez - Principal bundle structures among second order frame bundles).
