Let $D\subset \mathbb{R}^d$ with $d\geqslant 1$ be a bounded open connected Lipschitz set, $\eta\in (0,1)$, $p>0$, and $\alpha>0$. Then the paper (On comparability of integral forms written by Bartlomiej Dyda, 2006) proves in (13) (the following \eqref{1}) that there exists a constant $C=C(d,p,\alpha,\eta,D)>0$, \begin{equation}\label{1}\tag{*} \int_D\int_D\frac{|u(x)-u(y)|^p}{|x-y|^{d+\alpha}}\,\mathrm{d}y\mathrm{d}x \leqslant C\int_D\int_{B(x,\eta\mathrm{dist}(x,\partial D))}\frac{|u(x)-u(y)|^p}{|x-y|^{d+\alpha}}\,\mathrm{d}y\mathrm{d}x, \end{equation} where $B(x,\eta\mathrm{dist}(x,\partial D))$ is the ball centered at $x$ with radius $\eta\mathrm{dist}(x,\partial D)$.
My question: if $D$ is a cube (a very special bounded Lipschitz domain), can the constant $C$ in \eqref{1} be independent of $D$ (that is, the constant $C$ is universal for all cubes)?
