  In prior joint work with Lewis, we developed a theory of enriched set-valued
$P$-partitions to construct a $K$-theoretic generalization of the Hopf algebra
of peak quasisymmetric functions. Here, we situate this object in a diagram of
six Hopf algebras, providing a shifted version of the diagram of $K$-theoretic
combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to
describe new $K$-theoretic analogues of the classical peak algebra. We also
study the Hopf algebras generated by Ikeda and Naruse's $K$-theoretic Schur
$P$- and $Q$-functions, as well as their duals. Along the way, we derive
several product, coproduct, and antipode formulas and outline a number of open
problems and conjectures.
