Critical logarithmic higher-order Hardy inequality

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $N>k\geq 2$, and let $u\in C_c^k(B_R)$. Critical logarithmic Hardy conjecture. The inequality

$$\left(\prod_{j=1}^{k}\frac{jN-k}{N}\right)^{N/k}\int_{B_R}\frac{|u|^{N/k}}{|x|^N\left(\log\frac{R}{|x|}\right)^N}\,dx\leq\int_{B_R}|\nabla^k u|^{N/k}\,dx$$

holds. Moreover, the constant $\left(\prod_{j=1}^{k}(jN-k)/N\right)^{N/k}$ is optimal and is not attained by any nonzero $u$ for which the right-hand side is finite. This is the critical-exponent logarithmic counterpart of the proposed higher-order Hardy inequalities; the source gives no evidence that it has been proved or refuted.