Model structure for double ∞-categories compatible with the square construction

Establish a model structure on bisimplicial sets that presents double ∞-categories and interacts compatibly with the square construction that sends an (∞,2)-category to a double ∞-category, thereby resolving the conjectural existence of such a model structure in the ∞-categorical setting.

Background

The desired quasi-categorical enrichment for cosmological unstraightening on marked objects could be achieved via a model structure on s^+ as above. One pathway to such a model structure would be to develop a robust model structure for double ∞-categories on bisimplicial sets that is compatible with the square construction (which associates to an (∞,2)-category a double ∞-category).

This existence was conjectured by Gaitsgory and Rozenblyum. While a corresponding result has been proven for strict double categories, the ∞-categorical analogue remains unresolved, and its resolution would likely facilitate the sought quasi-categorical enrichment.