Maz’ya’s Problem 8 on the capacitary-distance Hardy inequality

Let $n\ge 3$. For every open set $\Omega\subset\mathbb{R}^n$, set $F:=\mathbb{R}^n\setminus\Omega$ and, for $0<\alpha\le 1$ and $x\in\Omega$, define $d_\alpha(x):=\inf\{r>0:\operatorname{cap}(\overline{F\cap B(x,r)})\ge\alpha\operatorname{cap}(B(\mathbf{0},r))\}$. Determine the optimal dependence on $\alpha$ of the least constant $C_n(\alpha)$ such that, for every such $\Omega$ and every $u\in C_c^\infty(\Omega)$, $\int_\Omega |u(x)|^2/d_\alpha(x)^2\,dx\le C_n(\alpha)\int_\Omega |\nabla u(x)|^2\,dx$. Equivalently, determine the sharp order of $C_n(\alpha)$ as $\alpha\downarrow0$; the claimed resolution is $C_n(\alpha)\asymp_n\alpha^{-2}$.