Model structures for cyclic algebras over cyclic ∞-operads

Determine whether model structures exist that model cyclic algebras for a cyclic ∞-operad, analogous to the covariant slice model structures on dendroidal sets or dendroidal spaces used to model algebras over an ∞-operad.

Background

For ∞-operads, covariant model structures on slices of dendroidal sets and dendroidal spaces provide homotopical models for algebras over a given ∞-operad, as developed in the work of Heuts and Moerdijk. Extending this paradigm to the cyclic context would give a homotopy-theoretic framework for cyclic algebras over cyclic ∞-operads.

The authors explicitly raise whether there are corresponding model structures in the cyclic setting, indicating an open direction to construct and characterize such model structures and to compare them to their non-cyclic counterparts.