Bar-construction compatibility for L^{1,1} and L^{2,1} localizations

Determine whether the L^{1,1}-localization, or alternatively the L^{2,1}-localization, commutes with the bar construction in the same way as the 0-birational localization.

Background

The paper explains that the 0-birational localization commutes with the bar construction, which eliminates the need to impose a separate strict birationality condition on framed motivic spaces in the n=0 case.

For the analogous L^{1,1} and L^{2,1} localizations, this compatibility is not known. Establishing it would clarify whether the corresponding localized recognition principles can be obtained without additional strict-locality hypotheses.