Morita equivalence of weakly and strictly stable weak identity types

Let $\mathbf{CwF}_{ws}$ be the category of contextual categories with weakly stable weak identity types, let $I_{ws}$ be the corresponding generating cofibrations, and let $L$ be the left adjoint in the free-forgetful adjunction between weakly stable and strictly stable structures. For an $I_{ws}$-cellular model $\mathcal{C}:\mathbf{CwF}_{ws}$, write $\eta:\mathcal{C}\to L(\mathcal{C})$ for the unit.

Morita equivalence conjecture. The theories of weakly stable weak identity types and strictly stable weak identity types are Morita equivalent: for every $I_{ws}$-cellular model $\mathcal{C}:\mathbf{CwF}_{ws}$, the unit

$$\eta:\mathcal{C}\to L(\mathcal{C})$$

is a weak equivalence.

This would provide the desired coherence theorem by showing that strictification preserves the homotopical theory of weakly stable identity types. The paper presents this as a goal toward a full coherence theorem, and no proof or resolution is supplied here.