Contractible p-component conjecture for the $(d-1)$-st Beer index

Let $d\geq 2$, let $S\subseteq\mathbb{R}^d$ be a set whose $(d-1)$-st Beer index $\operatorname{b}_{d-1}(S)$ is defined, and assume that every p-component of $S$ is contractible. Let $\operatorname{c}(S)$ denote the index of convexity. Contractible p-component conjecture. There is a constant $\alpha=\alpha(d)>0$ such that

$$\operatorname{b}_{d-1}(S)\leq\alpha\operatorname{c}(S).$$

This proposes a higher-dimensional analogue of the planar linear upper bound, replacing simple connectivity by contractibility of every p-component. The statement is presented as a possible generalization and remains open.