Equivalent image-classification form of the L'vov–Kaplansky conjecture

Let $K$ be an infinite field, let $n\in\mathbb{N}$, and let $p$ be a noncommutative multilinear polynomial evaluated on the matrix algebra $M_n(K)$. Let $K$ denote the set of scalar matrices, and let $\operatorname{sl}_n(K)$ denote the set of matrices with trace zero.

Equivalent L'vov–Kaplansky conjecture. The image of $p$ satisfies

$$\operatorname{Im}p\in\bigl\{\{0\},K,\operatorname{sl}_n(K),M_n(K)\bigr\}.$$

The source presents this as a reformulation of the L'vov–Kaplansky conjecture. It is known in some cases, notably $n=2$ over quadratically closed fields and over $\mathbb{R}$, whereas only partial results are reported for $n>2$.