Improved magnetic Hardy inequality with a critical remainder

Let $d\geq 2$, let $\nabla_A$ denote the magnetic gradient, let $h_{A,p}$ be the corresponding magnetic $p$-Laplacian form domain, and let $\mu_{p,d}$ be the Hardy constant. Let $B$ be a smooth closed magnetic field with $B\neq 0$, and let $A$ satisfy $dA=B$. Improved magnetic Hardy conjecture. For $2\leq p<d$, there exists a constant $c_{B,p,d}>0$ such that

$$\int_{\mathbb{R}^d}|\nabla_Au|^p\,dx-\mu_{p,d}\int_{\mathbb{R}^d}\frac{|u|^p}{|x|^p}\,dx\geq c_{B,p,d}\int_{\mathbb{R}^d}\left|\nabla_A\left(u|x|^{\frac{d-p}{p}}\right)\right|^p|x|^{p-d}\,dx,$$

for all $u\in\mathcal{D}(h_{A,p})$. The corresponding non-magnetic inequality is known, while the magnetic extension is proposed to show that a nontrivial magnetic field makes the critical operator subcritical; its status is not resolved in the supplied text.