Cyclic analogue of the Boardman–Vogt tensor product and dendroidal tensoring

Develop a cyclic analogue of the Boardman–Vogt tensor product for operads by defining a tensor product between anti-involutive simplicial sets and representable cyclic dendroidal sets #1{T}, in order to enable lifting of the second Quillen equivalence between dendroidal Rezk model structures and dendroidal sets to the cyclic setting.

Background

In the non-cyclic setting, Cisinski and Moerdijk constructed a Quillen equivalence between dendroidal Rezk model structures and dendroidal sets using the tensor product of dendroidal sets, which relies on the Boardman–Vogt tensor product of operads. The authors establish cyclic counterparts of several Quillen equivalences in this paper, but note that reproducing this particular equivalence in the cyclic context requires additional tensorial machinery.

Specifically, the missing ingredient is a tensor product operation between anti-involutive simplicial sets and representable cyclic dendroidal sets, which would serve as the cyclic analogue of the classical Boardman–Vogt tensor and the dendroidal tensor product used in the non-cyclic case. Defining such an operation would likely allow the lifting of the second Quillen equivalence to the cyclic framework.