Preservation of L^{n+1,n+1}-local equivalences by the framed forgetful functor

Prove that the framed-to-unframed functor $\gamma_*$ preserves $L^{n+1,n+1}$-local equivalences, not merely $L^{n+1,n+1}$-local objects.

Background

The framed reconstruction and recognition arguments use the functor $\gamma_*$ from framed motivic objects to ordinary motivic objects. The paper notes that preservation of $L^{n+1,n+1}$-local objects follows essentially from the definitions.

However, preservation of local equivalences is a stronger property and is required to ensure that $\gamma_*$ commutes with the corresponding localization functors. The authors cite an analogous result for n-birational localization but do not establish it for $L^{n+1,n+1}$.