L'vov-Kaplansky conjecture for multilinear noncommutative polynomials

Let $K$ be a field, let $m\in\mathbb{N}$, and let $K\langle x_1,\ldots,x_m\rangle$ be the free algebra in variables $x_1,\ldots,x_m$. A polynomial $f\in K\langle x_1,\ldots,x_m\rangle$ is multilinear if

$$f(x_1,\ldots,x_m)=\sum_{\sigma\in S_m}\lambda_\sigma x_{\sigma(1)}\cdots x_{\sigma(m)}$$

for some $\lambda_\sigma\in K$. For a $K$-algebra $A$, write

$$f(A)=\{f(a_1,\ldots,a_m)\mid a_1,\ldots,a_m\in A\}$$

for the image of $f$ in $A$.

L'vov–Kaplansky conjecture. If $f$ is multilinear, then its image in the matrix algebra $M_n(K)$ is a vector space for every $n\in\mathbb{N}$.

The conjecture was initiated by Kaplansky and later formulated by L'vov. The paper proves it when $f$ has degree $3$ and $K$ is an algebraically closed field of characteristic $0$; the general statement is therefore solved in that case but remains open in full generality.