Sharp higher-order Hardy inequality with distance to the boundary

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $1<p<\infty$, let $k\geq 2$, and let $u\in C_c^k(B_R)$. Write $\operatorname{dist}(x,\partial B_R)$ for the distance from $x$ to the boundary. Boundary-distance Hardy conjecture. The inequality

$$\left(\prod_{j=1}^{k}\frac{jp-1}{p}\right)^p\int_{B_R}\frac{|u|^p}{\operatorname{dist}(x,\partial B_R)^{kp}}\,dx\leq\int_{B_R}|\nabla^k u|^p\,dx$$

holds. Moreover, the constant $\left(\prod_{j=1}^{k}(jp-1)/p\right)^p$ is optimal and is not attained by any nonzero $u$ for which the right-hand side is finite. This is a proposed sharp higher-order Hardy inequality governed by distance to the boundary; the source does not provide a resolution.