Fermat's Last Theorem for integer matrices

For $k=2$ with $n \equiv 2 \mod 4$:

$$ \pmatrix{1 & (-1)^{(n-2)/4} 2^{n/2} n^{n-1} \cr 0 & 1}^n + \pmatrix{n & -n\cr n & n\cr}^n = \pmatrix{1 & 0\cr (-1)^{(n-2)/4} 2^{n/2} n^{n-1} & 1\cr}^n $$

An example for $n=4$ is $$ \pmatrix{3 & -2\cr 1 & 2\cr}^4 + \pmatrix{2 & -4\cr 2 & 0\cr}^4 = \pmatrix{1 & 2\cr -1 & 2\cr}^4 $$ but I don't have a generalization.

This problem is addressed in "On Fermat's problem in matrix rings and groups," by Z. Patay and A. Szakács, Publ. Math. Debrecen 61/3-4 (2002), 487–494, which summarizes previous work on the topic and gives some new results. It seems that the problem is not completely solved.

When $k=2$, Khazonov showed that there are solutions in $SL_2(\mathbb Z)$ if and only if $n$ is not a multiple of 3 or 4, but I couldn't immediately find any statement anywhere about the case $4\mid n$ and $2\times 2$ integer matrices with nonzero determinant.

Khazanov also proved that $GL_3(\mathbb Z)$ solutions do not exist if $n$ is a multiple of either 21 or 96, and $SL_3(\mathbb Z)$ solutions do not exist if $n$ is a multiple of 48.

Patay and Szakács give explicit solutions for $SL_3(\mathbb Z)$ when $n=\pm 1\pmod 3$ as well as for $n=3$. Here's a solution for $n=3$: $$\pmatrix{0& 0&1\\ 0 &-1& 1\\ 1 & 1 & 0}^3 + \pmatrix{0&1&0\\ 0&1&-1\\ -1&-1&0}^3 = \pmatrix{0&1&1\\0&0&1\\1&0&0}^3.$$