On plank coverings of cube annuli

We study plank coverings of cube annuli $Q\setminus \delta Q\subset {\mathbb{R}}^d$, where $Q$ is the $d$-dimensional unit cube and $\delta Q$ is a homothetic copy lying in its interior. A. Bezdek proved that, in dimension $2$, every plank covering of such an annulus has total width at least~1 whenever $\delta \le 1-1\mathrm{/}\sqrt{2}$. Smurov, Bogataya, and Bogatyi extended this result to all $d\ge 2$ under the condition $\delta \le 1-1\mathrm{/}\sqrt{d}$. It is conjectured that the same lower bound holds for $\delta \le 1-1\mathrm{/}d$. We provide evidence for this conjecture in dimensions $2$ and $3$ by proving that there exist positive constants ${\epsilon }_2$ and ${\epsilon }_3$ such that the bound remains valid whenever $\delta \le 1-1\mathrm{/}\sqrt{d}+{\epsilon }_d$.