Beilinson's conjecture on the level of spectra

I refer to the MO post here for a statement of Beilinson's conjectures about the Beilinson's regulator as a map from motivic cohomology: Beilinson conjectures

A result of Bunke-Nikolaus-Tamme shows that the regulator can be refined to a map of spectra. Has there been any formulation or even spectulation as to how the statement regarding the $$L$$-value in this setting?