Waring's problem over finite fields

Question

I am trying to answer the following question: Let $F_q$ be a finite field of characteristic $2$, and $M_2(F_q)$ the ring of $2 \times 2 $ matrices over $F_q$. I am able to see by computer programming that the matrix $\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}$ is not a sum of two $6$th powers. I wonder whether this is the only exception, i.e. for any $n > 2$, for any $k \neq 6$ and for any $q = 2$ is it true that any matrix in $M_n(F_2)$ is a sum of two $k$th powers? I know that eventually it is true, i.e. for sufficiently large $q$, every matrix in $M_n(F_q)$ for any $n$ is a sum of two $k$th powers. I also know that it is false for $n =2$ and for $n =3 $. For $n = 2$, the above matrix is one among three exception; this was found using computer. I wonder whether for $n \geq 4$ and $q = 2$ there won't be any exceptions.