Bezdek–Bezdek's successive inradius plank conjecture

Let $\mathbf{K}$ and $\mathbf{C}$ be convex bodies in $\mathbb{E}^d$, with $d\geq 2$, and let $m$ be a positive integer. For each body, let $r_{\mathbf{C}}(\cdot,m)$ denote its $m$th successive $\mathbf{C}$-inradius. Bezdek–Bezdek's successive inradius plank conjecture. If $\mathbf{K}$ is covered by the planks $\mathbf{P}_1,\mathbf{P}_2,\dots,\mathbf{P}_n$ in $\mathbb{E}^d$, then

$$\sum_{i=1}^n r_{\mathbf{C}}(\mathbf{P}_i,m)\geq r_{\mathbf{C}}(\mathbf{K},m),$$

or equivalently,

$$\sum_{i=1}^n w_{\mathbf{C}}(\mathbf{P}_i)\geq m r_{\mathbf{C}}(\mathbf{K},m).$$

This is stated as an equivalent strengthening of the affine plank and slicing conjectures, and its resolution is not supplied in the source context.