The graded L'vov–Kaplansky conjecture for matrix algebras of prime order

Let $K$ be a field and let $n$ be a prime number. Let $K\langle X\mid \mathbb{Z}_n\rangle$ denote the free $\mathbb{Z}_n$-graded algebra, and let $f\in K\langle X\mid \mathbb{Z}_n\rangle$ be a multilinear $\mathbb{Z}_n$-graded polynomial. For $g\in\mathbb{Z}_n$, write $M_n(K)_g$ for the homogeneous component of degree $g$ in the graded matrix algebra, and write $sl_n(K)_0$ for the degree-zero component of the trace-zero matrices. Graded L'vov–Kaplansky conjecture. The image $\operatorname{Im}(f)$ is one of

$$\{0\},\quad K,\quad sl_n(K)_0,\quad \text{or}\quad M_n(K)_g\text{ for some }g\in\mathbb{Z}_n.$$

The preceding argument proves the assertion over $K=\mathbb{Q}$, while the analogous statement for an arbitrary field remains open.