Unbounded commutator conjecture for self-adjoint operators

Let $A$ be an unbounded self-adjoint operator on a Hilbert space. Unbounded commutator conjecture. There is always a bounded self-adjoint operator $B$ such that $AB-BA$ is unbounded. This conjecture asks whether every unbounded self-adjoint operator has a bounded self-adjoint operator with which its commutator is unbounded; it is presented as an open question motivated by an earlier pair of operators in the paper.