Preservation of the arithmetic hierarchy for cohesiveness and Erdős–Moser principles

Let $coperatorname{COH}$ denote the cohesiveness principle and let $coperatorname{EM}$ denote the Erdős–Moser principle. A principle admits preservation of the arithmetic hierarchy if, for every instance and every set defined at an arithmetic level, there is a solution preserving that definability level.

Preservation conjecture. Both $coperatorname{COH}$ and $coperatorname{EM}$ admit preservation of the arithmetic hierarchy.

The paper has established preservation of $cDelta^0_2$-definitions and preservation beyond the $cDelta^0_2$ level for these principles separately, and conjectures that the two forms of preservation can be combined. The stated claim is not resolved in the supplied context.