Improved dimension dependence for approximate matchings

Let $M$ and $N$ be persistence modules that are $\epsilon$-interleaved, with $\epsilon\geq 0$. Let

$$r=\sup_{M'\in B(M)}\operatorname{supdim}M'<\infty.$$

Here $B(M)$ is the relevant neighborhood of $M$ defined in the paper, and an $\epsilon$-refinement is the paper's approximate decomposition relation. The improved-dimension conjecture. There is a constant $c\geq 0$, independent of $M$, $N$, $\epsilon$, and $r$, and a module $Q$ that is both a $cr\epsilon$-refinement of $M$ and a $cr\epsilon$-refinement of $N$.

This conjecture seeks to improve the main theorem's dependence on the maximum pointwise dimension of $M$ by using the dimensions of modules in $B(M)$. It is presented as open and is the main conjectural strengthening motivating later special cases.