Agrachev's universal gap conjecture for quantum control systems

Let $\Sigma$ be a finite-dimensional quantum control system on $\mathbb{C}^n$ of the form

$$\dot{x}=Ax+\sum_{i=1}^{p}B_i x\,u_i,$$

where $A$ and the $B_i$ are skew-adjoint matrices, and normalize the drift by $\lVert A\rVert=1$. Let $T(\Sigma)$ be the supremum, over pairs of points on the unit sphere $S^{2n-1}\subset\mathbb{C}^n$, of the minimum time needed to connect them using trajectories of $\Sigma$ with arbitrary $L^\infty$ controls. Agrachev's universal gap conjecture. There exists a universal gap $\delta>0$ for the minimum time: for every dimension $n$ and every quantum system $\Sigma$, either $T(\Sigma)=0$ or $T(\Sigma)\geq\delta$. The conjecture asserts a dimension-independent positive lower bound for every nonzero minimum controllability time, complementing the case in which the system without drift is controllable and $T(\Sigma)=0$.