Kaplansky–Lvov conjecture for multilinear polynomial images on matrix algebras

Let $p$ be a multilinear polynomial in $m$ variables over a field $k$, and let $\Phi_p$ be the associated polynomial map on $M_n(k)$. Write $\mathfrak{sl}_n(k)$ for the set of trace-zero matrices in $M_n(k)$. Kaplansky–Lvov conjecture. The image satisfies

$$\operatorname{Im}\Phi_p\in\bigl\{\{0\},\{aI:a\in k\},\mathfrak{sl}_n(k),M_n(k)\bigr\}.$$

The conjecture concerns the possible images of multilinear noncommutative polynomials on matrix algebras. The source says it was open for $n\geq 2$ at the time of writing, with the cases $n=2$ and $n=3$ having substantial partial results; it also notes that the conjecture was subsequently settled by Luo and Wang and independently by Gargate and de Mello.