Derived Interleaving Metric conjecture

Let $P$ be a poset and let $M$ and $N$ be persistence modules over $P$. Let $d_I(M,N)$ denote their interleaving distance, and let $d_{DG}(M,N)$ denote the derived convolution metric on the derived category, defined by the minimal thickening of the support needed for $M$ and $N$ to become isomorphic after convolution with a suitable kernel supported near the diagonal. Derived Interleaving Metric conjecture. There exist constants $C_1,C_2>0$, independent of $M$ and $N$, such that

$$C_1d_{DG}(M,N)\leq d_I(M,N)\leq C_2d_{DG}(M,N).$$

Thus the classical interleaving distance and the derived metric are bilipschitz equivalent. This proposes a quantitative comparison between abelian-category stability and derived deformation-theoretic stability; the source provides no proof or status evidence beyond presenting it as a conjecture.