Matching conjecture for upset-decomposable modules

Let $M$ and $N$ be upset decomposable persistence modules, where an upset decomposable module is a direct sum of upset modules. The upset matching conjecture. There is a constant $c\geq 1$ such that, for any $\epsilon$-interleaved upset decomposable modules $M$ and $N$, there is a $c\epsilon$-matching between $M$ and $N$.

This is the restriction of the paper's main conjecture to upset decomposable modules. The paper presents it as an open conjecture and relates it to the existence of common approximate refinements.