When [this survey][1] 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][2]. 

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


  [1]: http://ams.math.uni-bielefeld.de/mathscinet/search/publdoc.html?pg1=IID&s1=339774&vfpref=html&r=7&mx-pid=2061852
  [2]: http://dx.doi.org/10.1090/memo/0610