Characterization of non-empty energy spaces for continuous magnetic fields

Let $B\in C(\mathbb R^2)$ be a continuous magnetic field, and let $\mathcal E_B$ denote the energy space associated with $B$. The notation $B=\operatorname{curl} A'$ means that the magnetic field is the curl of a vector potential $A'$. Energy-space conjecture. The energy space $\mathcal E_B$ is non-empty if and only if there exists $A'\in L^2(\mathbb R^2;\mathbb R^2)$ such that

$$B=\operatorname{curl} A'.$$

The preceding theorems establish this characterization under additional regularity and boundedness assumptions on the magnetic field or potential; the conjecture proposes that continuity of $B$ alone suffices.