The persistent-homology lifespan upper-bound conjecture

Let $G$ be the filtered simplicial complex under consideration, let $\mathrm{PD}_k(G)$ denote the set of persistence pairs in degree $k$, and let $\Omega$ be a $k$-dimensional homology class with volume $|\Omega|$. For a pair $(b_\Omega,d_\Omega)\in\mathrm{PD}_k(G)$, its life span is $d_\Omega-b_\Omega$. Persistent-homology lifespan upper-bound conjecture. Let $(b_\Omega,d_\Omega)\in\mathrm{PD}_k(G)$. If $|\Omega|$ is the volume of the homology class $\Omega$, then

$$d_\Omega-b_\Omega\leq \sqrt[k]{|\Omega|}+1.$$

This conjecture generalizes the paper's upper bounds from surfaces to higher-dimensional homology classes.