The one-dimensional reduction conjecture for a magnetic barrier

Assume that $-1\leq a<0$ and $1/|a|<b<1/\beta_a$, where $\beta_a$ is defined by the spectral quantity in the paper, and let $\mathfrak e_a(b)$ be the energy introduced in the corresponding definition. For each $\xi\in\mathbb R$, let $E_{a,b}^{1D}(\xi)$ be the one-dimensional ground-state energy defined in the paper, and set

$$E_{a,b}^{1D}=\inf_{\xi\in\mathbb R}E_{a,b}^{1D}(\xi).$$

One-dimensional reduction conjecture. Under these assumptions,

$$\mathfrak e_a(b)=E_{a,b}^{1D}.$$

This conjectures that the energy associated with the step magnetic field is exactly captured by the corresponding one-dimensional minimization over the translational parameter $\xi$, extending the analogous reduction known for a uniform magnetic field. The parser provides no evidence that this conjecture has been resolved.