When the survey Axiomatic Stable Homotopy, Neil Strickland, 2004 was written the category of motivic spectra was not investigated from the point of view of axiomatic stable homotopy, as considered e.g. in Hovey, Palmieri and Strickland, Axiomatic Stable Homotopy Theory, 1997.

The question is this: was any work done in the last ten years in this direction?