Miller's Aharonov–Casher lower-bound conjecture

Let $B(x)\ge 0$ be a magnetic field with flux

$$\Phi:= \frac{1}{2\pi}\int B,$$

which may be infinite, and let $H$ denote the corresponding Pauli operator. Miller's conjecture. The dimension of its kernel should satisfy

$$\dim\operatorname{Ker}(H)\ge \lfloor\Phi\rfloor.$$

The conjecture was recalled from Miller, but the source states that it is false; the paper gives a counterexample to the Aharonov–Casher theorem for a continuous bounded field with infinite total variation, finite limiting flux greater than $1$, and trivial kernel.