Conjecture: Coarsening monotonicity of E′k

Demonstrate that the maximal-sum k-entanglement measure E′k is coarsening monotonic, i.e., it does not increase under coarsening of multipartite partitions of type (a).

Background

The authors establish a tight coarsening inequality for E′k but cannot conclude its coarsening monotonicity.

They explicitly conjecture E′k to be coarsening monotonic.

References

Although Eq.~\eref{m-coarsening} is valid for $E'_k$, we can not derive it is coarsening monotonic. In general, tightly coarsening monotonicity if stronger than the coarsening monotonicity, we thus conjecture that $E'_k$ is coarsening monotonic.

— Measure of entanglement and the monogamy relation: a topical review  (2512.21992 - Guo et al., 26 Dec 2025) in Section 11.1 The maximal sum