Aharonov–Bohm magnetic Hardy inequality for nonintegral flux

Let $d=2$, let $1<p<2$, and define the Aharonov–Bohm potential

$$A_\beta(x)=\beta\frac{(x_2,-x_1)}{|x|^2},\qquad \beta\in\mathbb{R}.$$

Aharonov–Bohm Hardy conjecture. If $\beta\notin\mathbb{Z}$, then there exists a constant $\lambda(p)>\left(\frac{2-p}{p}\right)^p$ such that

$$\int_{\mathbb{R}^2}|\nabla_{A_\beta}u|^p\,dx\geq\lambda(p)\int_{\mathbb{R}^2}\frac{|u|^p}{|x|^p}\,dx,$$

for all $u\in C_c^\infty(\mathbb{R}^2)$. This predicts a strict improvement over the nonmagnetic Hardy constant when the magnetic flux is nonintegral; the supplied text does not state whether it has been proved.