In G. M. L. Powell's note 'Steenrod operations in motivic cohomology', he stated that if $char(k)=0$, $$H^{*,*}(k,\mathbb{Z}/2)=K_*^M(k)/2[\tau]$$ where $\tau\in H^{0,1}$ is the unique nonzero element.

I wonder whether this result holds when $char(k)>0$?