Motivic Peterson conjecture

Let $M_n$ be the mod $2$ motivic cohomology algebra of the classifying space of the elementary abelian $2$-group of rank $n$, and let $QM_n^{d,*}$ denote its degree-$(d,*)$ quotient by the positive-degree action of the mod $2$ motivic Steenrod algebra. Let $\beta(d)$ be the number of ones in the binary expansion of $d$. Motivic Peterson conjecture. If $\beta(d)>n$, then

$$\dim QM_n^{d,*}=0.$$

This is the motivic analogue of the Peterson hit problem: it predicts that every element of degree $(d,*)$ is hit by a positive-degree motivic Steenrod operation when the binary weight of $d$ exceeds the rank $n$.