Positive controllability diameter conjecture for bilinear Schrödinger systems

Let $M$ be a connected complete Riemannian manifold. For a Hamiltonian of the form specified in the paper, let $T_q$ and $T_c$ denote the quantum and classical controllability diameters, respectively.

Controllability-diameter conjecture. For every such Hamiltonian, the implication

$$T_c>0 \quad\Rightarrow\quad T_q>0$$

holds.

This is a precise formulation of the expected obstruction to small-time controllability after quantization. The surrounding discussion motivates it through semiclassical approximations, but no resolution is given in the supplied text.