Towards coherence theorems for equational extensions of type theories

We study the conservativity of extensions by additional strict equalities of dependent type theories (and more general second-order generalized algebraic theories). The conservativity of Extensional Type Theory over Intensional Type Theory was proven by Hofmann. Our goal is to generalize such results to type theories without the Uniqueness of Identity Proofs principles, such as variants of Homotopy Type Theory. For this purpose, we construct what is essentially the infinity-congruence on the base theory that is freely generated by the considered equations. This induces a factorization of any equational extension, whose two factors can be studied independently. We conjecture that the first factor is always an equivalence when the base theory is well-behaved. We prove that the second factor is an equivalence when the infinity-congruence is 0-truncated.