Multiplicity-freeness conjecture for antipodes of the universal Hopf class

Let the universal Hopf class be denoted by $\textbf{\textsc{pre}}$. Multiplicity-freeness conjecture. The antipode for the Hopf monoid of the universal Hopf class $\textbf{\textsc{pre}}$ is multiplicity-free. This conjecture proposes a uniform cancellation property for antipodes across Hopf monoids in the universal class; the paper establishes multiplicity-free and cancellation-free formulas for important submonoids, but the universal assertion remains unresolved.