Polynomial growth conjecture for reachable sets

Let $\mathcal{X}$ be a real analytic control system that is small-time locally controllable from $x_0$. Write $\mathrm{R}_{\mathcal{X}}(<t,x_0)$ for the set reachable from $x_0$ in time less than $t$, and let $\overline{\mathrm{B}}(x_0,r)$ denote the closed ball of radius $r$ centered at $x_0$.

Polynomial growth conjecture. There exist $N\in\mathbb{N}$ and $T,C>0$ such that

$$\overline{\mathrm{B}}(x_0,Ct^N)\subseteq \mathrm{R}_{\mathcal{X}}(<t,x_0),\qquad \forall t\leq T.$$

This conjecture asserts a uniform polynomial lower bound on the size of reachable sets near the initial point. It is presented as a proposed characterization of the rate of growth associated with small-time local controllability; the source gives no resolution.