Lvov–Kaplansky conjecture on multilinear polynomial images of matrix algebras

Let $n\geq 2$ be an integer, let $K$ be a field, and let $K\langle X\rangle$ be the free associative algebra over $K$ on a countable set of noncommutative variables. A multilinear polynomial is an element of this free algebra that is multilinear in its variables. Lvov–Kaplansky conjecture. If $f$ is a multilinear polynomial, then the set of values of $f$ on the matrix algebra $M_n(K)$ is a vector space. The conjecture is a foundational problem about images of polynomial maps on matrix algebras; the source presents it as an old and famous conjecture, without giving a resolution status.