Finite-jet conjecture for small-time local controllability

Let $\mathcal{X}=\{X_0,X_1,\ldots,X_m\}$ be a real analytic control system on $\mathbb{R}^n$ that is small-time locally controllable from $x_0$. For another real analytic system $\mathcal{Y}=\{Y_0,Y_1,\ldots,Y_m\}$, suppose that there is an integer $N\in\mathbb{N}$ such that, for every $i\in\{0,1,2,\ldots,m\}$, the derivatives of $X_i$ and $Y_i$ at $x_0$ agree through order $N$:

$$D^{\mathbf r}X_i(x_0)=D^{\mathbf r}Y_i(x_0),\qquad \forall\mathbf r=(r_1,\ldots,r_n)\in\mathbb{Z}_{\geq 0}^n,\quad \sum_{j=1}^n r_j\leq N.$$

Finite-jet conjecture. There exists $N\in\mathbb{N}$ such that every such system $\mathcal{Y}$ is small-time locally controllable from $x_0$. The conjecture asks whether small-time local controllability of a real analytic system can be determined from finitely many derivatives of its vector fields at the initial point. No resolution is supplied in the source.