The width lower-bound conjecture for persistent-homology life spans

Let $G$ be the filtered simplicial complex under consideration, let $\mathrm{PD}_k(G)$ denote the degree-$k$ persistence pairs, and let $\Omega$ be the associated homology class with width $\omega(\Omega)$. For $(b_\Omega,d_\Omega)\in\mathrm{PD}_k(G)$, the life span is $d_\Omega-b_\Omega$. Width lower-bound conjecture. For any $k\geq 2$, there exists a constant $C_k<1$ such that, for every $(b_\Omega,d_\Omega)\in\mathrm{PD}_k(G)$,

$$C_k\cdot\omega(\Omega)\leq d_\Omega-b_\Omega.$$

The conjecture proposes a lower bound for persistence life spans in terms of the width of the represented homology class; the paper does not establish such lower bounds.