Nelson's essential skew-adjointness conjecture

Let $p=2$ and let $\mathbf{v}\in L^2(\mathbb{R}^d)$ be a compactly supported, divergence-free vector field. Define

$$\mathsf{A}(\rho)=\mathbf{v}\cdot\nabla\rho,\qquad D(\mathsf{A})=C_0^\infty(\mathbb{R}^d),$$

as an operator on $L^2(\mathbb{R}^d)$. Nelson's conjecture. The operator $\mathsf{A}$ is essentially skew-adjoint, equivalently

$$\mathsf{A}^*=-\overline{\mathsf{A}}.$$

Nelson formulated this in connection with existence and uniqueness of flows for vector fields with minimal regularity. The conjecture was later disproved in every dimension, including $d=2$, so it is refuted.