Multiplicity-freeness conjecture for antipodes of Hopf-class monoids

Let $\textbf{\textsc{pre}}^\natural$ denote the Hopf monoid arising from a Hopf class; equivalently, consider every Hopf monoid arising from a Hopf class. Multiplicity-freeness conjecture. In every Hopf monoid arising from a Hopf class (equivalently, in $\textbf{\textsc{pre}}^\natural$), the antipode is multiplicity-free. This is a broader formulation of the paper's antipode multiplicity-freeness question. The paper obtains such formulas for particular Hopf monoids, but the assertion for every Hopf monoid arising from a Hopf class remains open.