Shen's conjecture on gradient conditions for magnetic Schrödinger operators

Let $n\geq 3$, let ${\bf B}$ be the magnetic field, and consider the magnetic Schrödinger operator $L_{{\bf a},V}$ with potential $V$. Theorem 0.11 of Shen concerns a bound on the number of negative eigenvalues when the potential has a non-trivial negative part.

Shen's conjecture. Some condition on $\nabla {\bf B}$ is necessary to prove Theorem 0.11 of Shen.

The source presents this as Shen's remark that a condition on the gradient of the magnetic field seems necessary, while the specific condition previously used is more restrictive than one would hope. The conjectural necessity of such a condition is not resolved here.