Equality of the sufficient and admissible parameter sets for conditional entropies

Let $\mathcal{D}_{\rm bulk}$ and $\mathcal{D}_{-\infty}$ denote the parameter sets for which the corresponding conditional entropies are monotone under conditionally mixing channels. Let $\widetilde{\mathcal{D}}_{\rm bulk}$ and $\widetilde{\mathcal{D}}_{-\infty}$ denote the sets defined by the sufficient conditions in the paper. Parameter-set equality conjecture.

$$\widetilde{\mathcal{D}}_{\rm bulk}=\mathcal{D}_{\rm bulk}\quad\text{and}\quad\widetilde{\mathcal{D}}_{-\infty}=\mathcal{D}_{-\infty}.$$

The proposition establishes the corresponding inclusions and proves necessity under finite-integrability assumptions, giving a tight characterization for discrete measures with finite support. The conjectured equalities without those additional assumptions remain unresolved; the parser marks this candidate as disproved, but the supplied evidence only describes it as a conjecture and does not state a counterexample.