The conjecture that interleaving distance is optimal over any field

Let $k$ be the field with respect to which $n\mathbf{\textup{-}Mod}$ and the homology functors $H_i$ are defined. For $i\in\mathbb Z_{\geq 0}$, a pseudometric $d$ on $\operatorname{obj}^*(n\mathbf{\textup{-}Mod})$ is called $i$-optimal if it is $i$-stable and, for every other $i$-stable metric $d'$ on $\operatorname{obj}^*(n\mathbf{\textup{-}Mod})$, one has $d'\leq d$; for $i=0$, the definition requires $d'(M,N)\leq d(M,N)$ for all $M,N\in\operatorname{im}(H_0\circ F^S)$. The function $d_I$ denotes the interleaving distance.

Interleaving optimality conjecture. For any field $k$ and $i\in\mathbb Z_{\geq 0}$, $d_I$ is $i$-optimal.

The theorem preceding this conjecture proves the claim when $k=\mathbb Q$ or $k=\mathbb Z/p\mathbb Z$ for a prime $p$. The conjecture asks whether the same optimality holds over arbitrary fields; the paper notes that it would follow from the proposed extension of the geometric-lifting proposition.